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1.1 Real Numbers: Algebra Essentials

Learning objectives.

In this section, you will:

• Classify a real number as a natural, whole, integer, rational, or irrational number.
• Perform calculations using order of operations.
• Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
• Evaluate algebraic expressions.
• Simplify algebraic expressions.

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.

But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

Classifying a Real Number

The numbers we use for counting, or enumerating items, are the natural numbers : 1, 2, 3, 4, 5, and so on. We describe them in set notation as { 1 , 2 , 3 , ... } { 1 , 2 , 3 , ... } where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers . Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the opposites of the natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

The set of rational numbers is written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } . Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed as a terminating or repeating decimal. Any rational number can be represented as either:

• ⓐ a terminating decimal: 15 8 = 1.875 , 15 8 = 1.875 , or
• ⓑ a repeating decimal: 4 11 = 0.36363636 … = 0. 36 ¯ 4 11 = 0.36363636 … = 0. 36 ¯

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Writing Integers as Rational Numbers

Write each of the following as a rational number.

Write a fraction with the integer in the numerator and 1 in the denominator.

• ⓐ 7 = 7 1 7 = 7 1
• ⓑ 0 = 0 1 0 = 0 1
• ⓒ −8 = − 8 1 −8 = − 8 1

Identifying Rational Numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

• ⓐ − 5 7 − 5 7
• ⓑ 15 5 15 5
• ⓒ 13 25 13 25

Write each fraction as a decimal by dividing the numerator by the denominator.

• ⓐ − 5 7 = −0. 714285 ——— , − 5 7 = −0. 714285 ——— , a repeating decimal
• ⓑ 15 5 = 3 15 5 = 3 (or 3.0), a terminating decimal
• ⓒ 13 25 = 0.52 , 13 25 = 0.52 , a terminating decimal
• ⓐ 68 17 68 17
• ⓑ 8 13 8 13
• ⓒ − 17 20 − 17 20

Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even 3 2 , 3 2 , but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers . Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

• ⓑ 33 9 33 9
• ⓓ 17 34 17 34
• ⓔ 0.3033033303333 … 0.3033033303333 …
• ⓐ 25 : 25 : This can be simplified as 25 = 5. 25 = 5. Therefore, 25 25 is rational.

So, 33 9 33 9 is rational and a repeating decimal.

• ⓒ 11 : 11 11 : 11 is irrational because 11 is not a perfect square and 11 11 cannot be expressed as a fraction.

So, 17 34 17 34 is rational and a terminating decimal.

• ⓔ 0.3033033303333 … 0.3033033303333 … is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
• ⓐ 7 77 7 77
• ⓒ 4.27027002700027 … 4.27027002700027 …
• ⓓ 91 13 91 13

Real Numbers

Given any number n , we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers . As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in Figure 1 .

Classifying Real Numbers

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

• ⓐ − 10 3 − 10 3
• ⓒ − 289 − 289
• ⓓ −6 π −6 π
• ⓔ 0.615384615384 … 0.615384615384 …
• ⓐ − 10 3 − 10 3 is negative and rational. It lies to the left of 0 on the number line.
• ⓑ 5 5 is positive and irrational. It lies to the right of 0.
• ⓒ − 289 = − 17 2 = −17 − 289 = − 17 2 = −17 is negative and rational. It lies to the left of 0.
• ⓓ −6 π −6 π is negative and irrational. It lies to the left of 0.
• ⓔ 0.615384615384 … 0.615384615384 … is a repeating decimal so it is rational and positive. It lies to the right of 0.
• ⓑ −11.411411411 … −11.411411411 …
• ⓒ 47 19 47 19
• ⓓ − 5 2 − 5 2
• ⓔ 6.210735 6.210735

Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 2 .

Sets of Numbers

The set of natural numbers includes the numbers used for counting: { 1 , 2 , 3 , ... } . { 1 , 2 , 3 , ... } .

The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the negative natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } .

The set of rational numbers includes fractions written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } .

The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: { h | h is not a rational number } . { h | h is not a rational number } .

Differentiating the Sets of Numbers

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

• ⓔ 3.2121121112 … 3.2121121112 …

• ⓐ − 35 7 − 35 7
• ⓔ 4.763763763 … 4.763763763 …

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example, 4 2 = 4 ⋅ 4 = 16. 4 2 = 4 ⋅ 4 = 16. We can raise any number to any power. In general, the exponential notation a n a n means that the number or variable a a is used as a factor n n times.

In this notation, a n a n is read as the n th power of a , a , or a a to the n n where a a is called the base and n n is called the exponent . A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, 24 + 6 ⋅ 2 3 − 4 2 24 + 6 ⋅ 2 3 − 4 2 is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations . This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify 4 2 4 2 as 16.

Next, perform multiplication or division, left to right.

Lastly, perform addition or subtraction, left to right.

Therefore, 24 + 6 ⋅ 2 3 − 4 2 = 12. 24 + 6 ⋅ 2 3 − 4 2 = 12.

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

Order of Operations

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :

P (arentheses) E (xponents) M (ultiplication) and D (ivision) A (ddition) and S (ubtraction)

Given a mathematical expression, simplify it using the order of operations.

• Step 1. Simplify any expressions within grouping symbols.
• Step 2. Simplify any expressions containing exponents or radicals.
• Step 3. Perform any multiplication and division in order, from left to right.
• Step 4. Perform any addition and subtraction in order, from left to right.

Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

• ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 )
• ⓑ 5 2 − 4 7 − 11 − 2 5 2 − 4 7 − 11 − 2
• ⓒ 6 − | 5 − 8 | + 3 ( 4 − 1 ) 6 − | 5 − 8 | + 3 ( 4 − 1 )
• ⓓ 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2
• ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1
• ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction. ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction.

Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.

• ⓒ 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition. 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition.

In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

• ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add. 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add.
• ⓐ 5 2 − 4 2 + 7 ( 5 − 4 ) 2 5 2 − 4 2 + 7 ( 5 − 4 ) 2
• ⓑ 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6
• ⓒ | 1.8 − 4.3 | + 0.4 15 + 10 | 1.8 − 4.3 | + 0.4 15 + 10
• ⓓ 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2
• ⓔ [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 ) [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 )

Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

Again, consider an example with real numbers.

It is important to note that neither subtraction nor division is commutative. For example, 17 − 5 17 − 5 is not the same as 5 − 17. 5 − 17. Similarly, 20 ÷ 5 ≠ 5 ÷ 20. 20 ÷ 5 ≠ 5 ÷ 20.

Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

Consider this example.

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

This property can be especially helpful when dealing with negative integers. Consider this example.

Are subtraction and division associative? Review these examples.

As we can see, neither subtraction nor division is associative.

Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

A special case of the distributive property occurs when a sum of terms is subtracted.

For example, consider the difference 12 − ( 5 + 3 ) . 12 − ( 5 + 3 ) . We can rewrite the difference of the two terms 12 and ( 5 + 3 ) ( 5 + 3 ) by turning the subtraction expression into addition of the opposite. So instead of subtracting ( 5 + 3 ) , ( 5 + 3 ) , we add the opposite.

Now, distribute −1 −1 and simplify the result.

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

For example, we have ( −6 ) + 0 = −6 ( −6 ) + 0 = −6 and 23 ⋅ 1 = 23. 23 ⋅ 1 = 23. There are no exceptions for these properties; they work for every real number, including 0 and 1.

Inverse Properties

The inverse property of addition states that, for every real number a , there is a unique number, called the additive inverse (or opposite), denoted by (− a ), that, when added to the original number, results in the additive identity, 0.

For example, if a = −8 , a = −8 , the additive inverse is 8, since ( −8 ) + 8 = 0. ( −8 ) + 8 = 0.

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a , there is a unique number, called the multiplicative inverse (or reciprocal), denoted 1 a , 1 a , that, when multiplied by the original number, results in the multiplicative identity, 1.

For example, if a = − 2 3 , a = − 2 3 , the reciprocal, denoted 1 a , 1 a , is − 3 2 − 3 2 because

Properties of Real Numbers

The following properties hold for real numbers a , b , and c .

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

• ⓐ 3 ⋅ 6 + 3 ⋅ 4 3 ⋅ 6 + 3 ⋅ 4
• ⓑ ( 5 + 8 ) + ( −8 ) ( 5 + 8 ) + ( −8 )
• ⓒ 6 − ( 15 + 9 ) 6 − ( 15 + 9 )
• ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) 4 7 ⋅ ( 2 3 ⋅ 7 4 )
• ⓔ 100 ⋅ [ 0.75 + ( −2.38 ) ] 100 ⋅ [ 0.75 + ( −2.38 ) ]
• ⓐ 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify. 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify.
• ⓑ ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition. ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition.
• ⓒ 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify. 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify.
• ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication. 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication.
• ⓔ 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify. 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify.
• ⓐ ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ] ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ]
• ⓑ 5 ⋅ ( 6.2 + 0.4 ) 5 ⋅ ( 6.2 + 0.4 )
• ⓒ 18 − ( 7 −15 ) 18 − ( 7 −15 )
• ⓓ 17 18 + [ 4 9 + ( − 17 18 ) ] 17 18 + [ 4 9 + ( − 17 18 ) ]
• ⓔ 6 ⋅ ( −3 ) + 6 ⋅ 3 6 ⋅ ( −3 ) + 6 ⋅ 3

Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as x + 5 , 4 3 π r 3 , x + 5 , 4 3 π r 3 , or 2 m 3 n 2 . 2 m 3 n 2 . In the expression x + 5 , x + 5 , 5 is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

• ⓑ 4 3 π r 3 4 3 π r 3
• ⓒ 2 m 3 n 2 2 m 3 n 2

• ⓐ 2 π r ( r + h ) 2 π r ( r + h )
• ⓑ 2( L + W )
• ⓒ 4 y 3 + y 4 y 3 + y

Evaluating an Algebraic Expression at Different Values

Evaluate the expression 2 x − 7 2 x − 7 for each value for x.

• ⓐ x = 0 x = 0
• ⓑ x = 1 x = 1
• ⓒ x = 1 2 x = 1 2
• ⓓ x = −4 x = −4
• ⓐ Substitute 0 for x . x . 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7
• ⓑ Substitute 1 for x . x . 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5
• ⓒ Substitute 1 2 1 2 for x . x . 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6
• ⓓ Substitute −4 −4 for x . x . 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15

Evaluate the expression 11 − 3 y 11 − 3 y for each value for y.

• ⓐ y = 2 y = 2
• ⓑ y = 0 y = 0
• ⓒ y = 2 3 y = 2 3
• ⓓ y = −5 y = −5

Evaluate each expression for the given values.

• ⓐ x + 5 x + 5 for x = −5 x = −5
• ⓑ t 2 t −1 t 2 t −1 for t = 10 t = 10
• ⓒ 4 3 π r 3 4 3 π r 3 for r = 5 r = 5
• ⓓ a + a b + b a + a b + b for a = 11 , b = −8 a = 11 , b = −8
• ⓔ 2 m 3 n 2 2 m 3 n 2 for m = 2 , n = 3 m = 2 , n = 3
• ⓐ Substitute −5 −5 for x . x . x + 5 = ( −5 ) + 5 = 0 x + 5 = ( −5 ) + 5 = 0
• ⓑ Substitute 10 for t . t . t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19 t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19
• ⓒ Substitute 5 for r . r . 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π
• ⓓ Substitute 11 for a a and –8 for b . b . a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85 a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85
• ⓔ Substitute 2 for m m and 3 for n . n . 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12
• ⓐ y + 3 y − 3 y + 3 y − 3 for y = 5 y = 5
• ⓑ 7 − 2 t 7 − 2 t for t = −2 t = −2
• ⓒ 1 3 π r 2 1 3 π r 2 for r = 11 r = 11
• ⓓ ( p 2 q ) 3 ( p 2 q ) 3 for p = −2 , q = 3 p = −2 , q = 3
• ⓔ 4 ( m − n ) − 5 ( n − m ) 4 ( m − n ) − 5 ( n − m ) for m = 2 3 , n = 1 3 m = 2 3 , n = 1 3

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation 2 x + 1 = 7 2 x + 1 = 7 has the solution of 3 because when we substitute 3 for x x in the equation, we obtain the true statement 2 ( 3 ) + 1 = 7. 2 ( 3 ) + 1 = 7.

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area A A of a circle in terms of the radius r r of the circle: A = π r 2 . A = π r 2 . For any value of r , r , the area A A can be found by evaluating the expression π r 2 . π r 2 .

Using a Formula

A right circular cylinder with radius r r and height h h has the surface area S S (in square units) given by the formula S = 2 π r ( r + h ) . S = 2 π r ( r + h ) . See Figure 3 . Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of π . π .

Evaluate the expression 2 π r ( r + h ) 2 π r ( r + h ) for r = 6 r = 6 and h = 9. h = 9.

The surface area is 180 π 180 π square inches.

A photograph with length L and width W is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm 2 ) is found to be A = ( L + 16 ) ( W + 16 ) − L ⋅ W . A = ( L + 16 ) ( W + 16 ) − L ⋅ W . See Figure 4 . Find the area of a mat for a photograph with length 32 cm and width 24 cm.

Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

Simplify each algebraic expression.

• ⓐ 3 x − 2 y + x − 3 y − 7 3 x − 2 y + x − 3 y − 7
• ⓑ 2 r − 5 ( 3 − r ) + 4 2 r − 5 ( 3 − r ) + 4
• ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) ( 4 t − 5 4 s ) − ( 2 3 t + 2 s )
• ⓓ 2 m n − 5 m + 3 m n + n 2 m n − 5 m + 3 m n + n
• ⓐ 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify. 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify.
• ⓑ 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify. 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify.
• ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify. ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify.
• ⓓ 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify. 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify.
• ⓐ 2 3 y − 2 ( 4 3 y + z ) 2 3 y − 2 ( 4 3 y + z )
• ⓑ 5 t − 2 − 3 t + 1 5 t − 2 − 3 t + 1
• ⓒ 4 p ( q − 1 ) + q ( 1 − p ) 4 p ( q − 1 ) + q ( 1 − p )
• ⓓ 9 r − ( s + 2 r ) + ( 6 − s ) 9 r − ( s + 2 r ) + ( 6 − s )

Simplifying a Formula

A rectangle with length L L and width W W has a perimeter P P given by P = L + W + L + W . P = L + W + L + W . Simplify this expression.

If the amount P P is deposited into an account paying simple interest r r for time t , t , the total value of the deposit A A is given by A = P + P r t . A = P + P r t . Simplify the expression. (This formula will be explored in more detail later in the course.)

Access these online resources for additional instruction and practice with real numbers.

• Simplify an Expression.
• Evaluate an Expression 1.
• Evaluate an Expression 2.

1.1 Section Exercises

Is 2 2 an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?

What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

For the following exercises, simplify the given expression.

10 + 2 × ( 5 − 3 ) 10 + 2 × ( 5 − 3 )

6 ÷ 2 − ( 81 ÷ 3 2 ) 6 ÷ 2 − ( 81 ÷ 3 2 )

18 + ( 6 − 8 ) 3 18 + ( 6 − 8 ) 3

−2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2 −2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2

4 − 6 + 2 × 7 4 − 6 + 2 × 7

3 ( 5 − 8 ) 3 ( 5 − 8 )

4 + 6 − 10 ÷ 2 4 + 6 − 10 ÷ 2

12 ÷ ( 36 ÷ 9 ) + 6 12 ÷ ( 36 ÷ 9 ) + 6

( 4 + 5 ) 2 ÷ 3 ( 4 + 5 ) 2 ÷ 3

3 − 12 × 2 + 19 3 − 12 × 2 + 19

2 + 8 × 7 ÷ 4 2 + 8 × 7 ÷ 4

5 + ( 6 + 4 ) − 11 5 + ( 6 + 4 ) − 11

9 − 18 ÷ 3 2 9 − 18 ÷ 3 2

14 × 3 ÷ 7 − 6 14 × 3 ÷ 7 − 6

9 − ( 3 + 11 ) × 2 9 − ( 3 + 11 ) × 2

6 + 2 × 2 − 1 6 + 2 × 2 − 1

64 ÷ ( 8 + 4 × 2 ) 64 ÷ ( 8 + 4 × 2 )

9 + 4 ( 2 2 ) 9 + 4 ( 2 2 )

( 12 ÷ 3 × 3 ) 2 ( 12 ÷ 3 × 3 ) 2

25 ÷ 5 2 − 7 25 ÷ 5 2 − 7

( 15 − 7 ) × ( 3 − 7 ) ( 15 − 7 ) × ( 3 − 7 )

2 × 4 − 9 ( −1 ) 2 × 4 − 9 ( −1 )

4 2 − 25 × 1 5 4 2 − 25 × 1 5

12 ( 3 − 1 ) ÷ 6 12 ( 3 − 1 ) ÷ 6

For the following exercises, evaluate the expression using the given value of the variable.

8 ( x + 3 ) – 64 8 ( x + 3 ) – 64 for x = 2 x = 2

4 y + 8 – 2 y 4 y + 8 – 2 y for y = 3 y = 3

( 11 a + 3 ) − 18 a + 4 ( 11 a + 3 ) − 18 a + 4 for a = –2 a = –2

4 z − 2 z ( 1 + 4 ) – 36 4 z − 2 z ( 1 + 4 ) – 36 for z = 5 z = 5

4 y ( 7 − 2 ) 2 + 200 4 y ( 7 − 2 ) 2 + 200 for y = –2 y = –2

− ( 2 x ) 2 + 1 + 3 − ( 2 x ) 2 + 1 + 3 for x = 2 x = 2

For the 8 ( 2 + 4 ) − 15 b + b 8 ( 2 + 4 ) − 15 b + b for b = –3 b = –3

2 ( 11 c − 4 ) – 36 2 ( 11 c − 4 ) – 36 for c = 0 c = 0

4 ( 3 − 1 ) x – 4 4 ( 3 − 1 ) x – 4 for x = 10 x = 10

1 4 ( 8 w − 4 2 ) 1 4 ( 8 w − 4 2 ) for w = 1 w = 1

For the following exercises, simplify the expression.

4 x + x ( 13 − 7 ) 4 x + x ( 13 − 7 )

2 y − ( 4 ) 2 y − 11 2 y − ( 4 ) 2 y − 11

a 2 3 ( 64 ) − 12 a ÷ 6 a 2 3 ( 64 ) − 12 a ÷ 6

8 b − 4 b ( 3 ) + 1 8 b − 4 b ( 3 ) + 1

5 l ÷ 3 l × ( 9 − 6 ) 5 l ÷ 3 l × ( 9 − 6 )

7 z − 3 + z × 6 2 7 z − 3 + z × 6 2

4 × 3 + 18 x ÷ 9 − 12 4 × 3 + 18 x ÷ 9 − 12

9 ( y + 8 ) − 27 9 ( y + 8 ) − 27

( 9 6 t − 4 ) 2 ( 9 6 t − 4 ) 2

6 + 12 b − 3 × 6 b 6 + 12 b − 3 × 6 b

18 y − 2 ( 1 + 7 y ) 18 y − 2 ( 1 + 7 y )

( 4 9 ) 2 × 27 x ( 4 9 ) 2 × 27 x

8 ( 3 − m ) + 1 ( − 8 ) 8 ( 3 − m ) + 1 ( − 8 )

9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x ) 9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x )

5 2 − 4 ( 3 x ) 5 2 − 4 ( 3 x )

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.

Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

How much money does Fred keep?

For the following exercises, solve the given problem.

According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by π . π . Is the circumference of a quarter a whole number, a rational number, or an irrational number?

Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

For the following exercises, consider this scenario: There is a mound of g g pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

Write the equation that describes the situation.

Solve for g .

For the following exercise, solve the given problem.

Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, then the department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. They must spend the budget such that 2,500,000 − x = 0. 2,500,000 − x = 0. What property of addition tells us what the value of x must be?

For the following exercises, use a graphing calculator to solve for x . Round the answers to the nearest hundredth.

0.5 ( 12.3 ) 2 − 48 x = 3 5 0.5 ( 12.3 ) 2 − 48 x = 3 5

( 0.25 − 0.75 ) 2 x − 7.2 = 9.9 ( 0.25 − 0.75 ) 2 x − 7.2 = 9.9

If a whole number is not a natural number, what must the number be?

Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.

Determine whether the simplified expression is rational or irrational: −18 − 4 ( 5 ) ( −1 ) . −18 − 4 ( 5 ) ( −1 ) .

Determine whether the simplified expression is rational or irrational: −16 + 4 ( 5 ) + 5 . −16 + 4 ( 5 ) + 5 .

The division of two natural numbers will always result in what type of number?

What property of real numbers would simplify the following expression: 4 + 7 ( x − 1 ) ? 4 + 7 ( x − 1 ) ?

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• Book title: College Algebra 2e
• Publication date: Dec 21, 2021
• Location: Houston, Texas
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Home / Boards / CBSE / Important Questions / Class 10 / Maths / Real Number

Class 10 Maths Chapter 1 Real Numbers Important Questions

Table of Contents

Introduction

What are real numbers.

• Real numbers consist of both rational and irrational numbers combined.
• Every real number can be represented on the number line.

Class 10 Real Numbers Important Questions and Answers

Q 1. the sum of exponents of prime factors in the prime-factorisation of 196 is:, (a) 3 (b) 4 (c) 5 (d) 2.

Explanation: Prime factorization is the method of representing a number as the multiplication of its prime factors. A number qualifies as prime if it possesses solely two factors , 1 and the number itself.Here's a breakdown of the sequential procedure for identifying the prime factors in the prime factorization of 196: Prime factors of 196 = 22 × 72 ‍So, sum of exponents = 2+2=4

Q 2. Complete the missing entries in the following factor tree:

(a) 42 and 21 (b) 24 and 12 (c) 7 and 3 (d) 84 and 42

Explanation: Using the factor tree, we have:

From the above factor tree. It is clear that               y = 3 ×7 = 21 and       x = 2 × y = 2 × 21                  = 42.

Q 3. The LCM of two numbers is 14 times their HCF. The sum of LCM and HCF is 750. If one number is 250, then find the other number.

Hence, the other number is 140.

Q 4. On a morning walk, three men step off together and their steps measure 54 cm, 60 cm and 48 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

Q 5. the lcm of two numbers is m times their hcf. the sum of lcm and hcf is 750. if one number is 250, then find the other number..

Ans. 140 Explanation: Let HCF be ‘H’   Then LCM =14H  sum of LCM and HCF is 750.  ∴  14H + H  = 750   ⇒ 15H = 750   \Rightarrow\text{H} =\large\frac{750}{15}  ⇒ H = 50 ∴ LCM = 14H   = 14×50 = 700 We know                 Product of LCM and HCF = Product of two numbers. Let other number be y Then,            700 × 50 = 250 × y   ⇒  \text{y} = \large\frac{700 × 50}{250} ⇒ y = 140 Hence, the other number is 140.

CBSE Class 10 Maths Chapter wise Important Questions

Frequently Asked Questions

Q1. what is the difference between rational and irrational numbers, q2. can real numbers be negative, q3. are all integers real numbers, q4. what is the decimal representation of a real number, q5. are all square roots irrational.

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• real-numbers-quiz

Real Numbers Quiz

Real numbers - practice problems with solutions.

Real Numbers

These lessons, with videos, examples and solutions, explain what real numbers are and some of their properties.

Related Pages The Real Number System Properties Of Real Numbers More Lessons for SAT Math Math Worksheets

The following diagram shows real numbers are made up of rational numbers, integers, whole numbers, and irrational numbers. Scroll down the page for more examples and solutions on real numbers and their properties.

Introduction to Real Numbers When analyzing data, graphing equations and performing computations, we are most often working with real numbers. Real numbers are the set of all numbers that can be expressed as a decimal or that are on the number line. Real numbers have certain properties and different classifications, including natural, whole, integers, rational and irrational.

This video goes over the basics of the real number system that is mainly used in Algebra. The video covers rational numbers, and irrational numbers.

Properties of Real Numbers When analyzing data or solving problems with real numbers, it can be helpful to understand the properties of real numbers. These properties of real numbers, including the Associative, Commutative, Multiplicative and Additive Identity, Multiplicative and Additive Inverse, and Distributive Properties, can be used not only in proofs, but in understanding how to manipulate and solve equations.

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Chapter 1 Class 10 Real Numbers

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Updated for NCERT 2023-2024 Book.

Answers to all exercise questions and examples are solved for Chapter 1 Class 10 Real numbers. Solutions of all these NCERT Questions are explained in a step-by-step easy to understand manner 

In this chapter, we will study

• What is a Real Number
• What is Euclid's Division Lemma , and
• How to find HCF (Highest Common Factor) using Euclid's Division Algorithm
• Then, we study Fundamental Theorem of Arithmetic, which is basically Prime Factorisation
• And find HCF and LCM using Prime Factorisation
• We also use the formula of HCF and LCM of two numbers a and b HCF × LCM = a × b
• Then, we see what is an Irrational Number
• and Prove numbers irrational (Like Prove  √ 2, √ 3 irrational)
• We revise our concepts about Decimal Expansion (Terminating, Non-Terminating Repeating, Non Terminating Non Repeating)
• And find out Decimal Expansion of numbers without performing long division

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Real Numbers

Real numbers are just numbers like:.

Nearly any number you can think of is a Real Number

Real Numbers include:

Real Numbers can also be positive , negative or zero .

So ... what is NOT a Real Number?

Mathematicians also play with some special numbers that aren't Real Numbers.

The Real Number Line

The Real Number Line is like a geometric line .

A point is chosen on the line to be the "origin" . Points to the right are positive, and points to the left are negative.

A distance is chosen to be "1", then whole numbers are marked off: {1,2,3,...}, and also in the negative direction: {...,−3,−2,−1}

Any point on the line is a Real Number:

• The numbers could be whole (like 7)
• or rational (like 20/9)
• or irrational (like π )

But we won't find Infinity, or an Imaginary Number.

Any Number of Digits

A Real Number can have any number of digits either side of the decimal point

• 0.000 000 0001

There can be an infinite number of digits, such as 1 3 = 0.333...

Why are they called "Real" Numbers?

Because they are not Imaginary Numbers

The Real Numbers had no name before Imaginary Numbers were thought of. They got called "Real" because they were not Imaginary. That is the actual answer!

Real does not mean they are in the real world

They are not called "Real" because they show the value of something real .

In mathematics we like our numbers pure, when we write 0.5 we mean exactly half.

But in the real world half may not be exact (try cutting an apple exactly in half).

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Classification of Real Numbers: Problems with Solutions

Natural numbers: $N=\left\{ 1,2,3,4,5,6,7,............\right\} $

Integers(whole numbers): $Z=\left\{......-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7.......\right\}$

Rational numbers: $Q=\left\{ ......-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7......\frac{3}{2},-\frac{1}{5},\frac{4}{3},....\right\}$

Irrational numbers: $I=\left\{ \sqrt{2},-\sqrt{5},e,\pi ....\right\} $. In the image R\Q represents the set of irrational numbers.

Real numbers: $R=Q\cup I$ - the real numbers are the union of rational and irrational.

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• Class 10 Maths
• Important Questions for Class 10 Maths Chapter 1- Real Numbers

Important Questions Class 10 Maths Chapter 1 Real Numbers

Important Class 10 maths questions for Chapter 1 real numbers are provided here to help the students practise better and score well in their CBSE Class 10 maths exam 2022-23. Additional questions of  real numbers  given here are as per the NCERT book. It covers the complete syllabus and will help the students to develop confidence and problem-solving skills for their exams.

Students can practise solving these questions after they have covered the syllabus for this chapter. Solving different types of questions will help them to clear their doubts. Let us see some questions here with their solutions. Get the important questions of Maths Class 10 for all the chapters here. Solving these important questions will boost the confidence level of students. They will also understand the difficulty level of the questions expected to be asked from this chapter.

Students can also get access to Class 10 Maths Chapter 1 Real Numbers MCQs here.

Before we move ahead with important questions let us learn about real numbers.

What are real numbers?

Real numbers are a combination of rational and irrational numbers. They are the numbers upon which we easily perform mathematical operations. All the numbers which are not imaginary are real numbers. For example, 22, -11, 7.99, 3/2, π(3.14), √2, etc.

Also Check:

• Important 2 Marks Questions for CBSE 10th Maths
• Important 3 Marks Questions for CBSE 10th Maths
• Important 4 Marks Questions for CBSE 10th Maths

Class 10 Real Numbers Important Questions and Answers

Q.1: Use Euclid’s division lemma to show that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

Let x be any positive integer and y = 3.

By Euclid’s division algorithm;

x =3q + r (for some integer q ≥ 0 and r = 0, 1, 2 as r ≥ 0 and r < 3)

x = 3q, 3q + 1 and 3q + 2

As per the given question, if we take the square on both the sides, we get;

x 2 = (3q) 2 = 9q 2 = 3.3q 2

Let 3q 2 = m

x 2 = 3m ………………….(1)

x 2 = (3q + 1) 2

= (3q) 2 + 1 2 + 2 × 3q × 1

= 9q 2 + 1 + 6q

= 3(3q 2 + 2q) + 1

Substituting 3q 2 +2q = m we get,

x 2 = 3m + 1 ……………………………. (2)

x 2 = (3q + 2) 2

= (3q) 2  + 2 2  + 2 × 3q × 2

= 9q 2 + 4 + 12q

= 3(3q 2 + 4q + 1) + 1

Again, substituting 3q 2  + 4q + 1 = m, we get,

x 2 = 3m + 1…………………………… (3)

Hence, from eq. 1, 2 and 3, we conclude that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

Q.2: Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429 Solution: (i) 140 Using the division of a number by prime numbers method, we can get the product of prime factors of 140. Therefore, 140 = 2 × 2 × 5 × 7 × 1 = 2 2  × 5 × 7

(ii) 156 Using the division of a number by prime numbers method, we can get the product of prime factors of 156.

Hence, 156 = 2 × 2 × 13 × 3 = 2 2 × 13 × 3

(iii) 3825 Using the division of a number by prime numbers method, we can get the product of prime factors of 3825.

Hence, 3825 = 3 × 3 × 5 × 5 × 17 = 3 2 × 5 2  × 17

(iv) 5005 Using the division of a number by prime numbers method, we can get the product of prime factors of 5005.

Hence, 5005 = 5 × 7 × 11 × 13 = 5 × 7 × 11 × 13

(v) 7429 Using the division of a number by prime numbers method, we can get the product of prime factors of 7429.

Hence, 7429 = 17 × 19 × 23 = 17 × 19 × 23

Q.3: Given that HCF (306, 657) = 9, find LCM (306, 657).

As we know that,

HCF × LCM = Product of the two given numbers

9 × LCM = 306 × 657

LCM = (306 × 657)/9 = 22338

Therefore, LCM(306,657) = 22338

Q.4: Prove that 3 + 2√5 is irrational.

Let 3 + 2 √ 5 be a rational number.

Then the co-primes x and y of the given rational number where (y ≠ 0) is such that:

3 + 2 √ 5 = x/y

Rearranging, we get,

2 √ 5 = (x/y) – 3

Since x and y are integers, thus, 1/2[(x/y) – 3] is a rational number.

Therefore, √ 5 is also a rational number. But this confronts the fact that √ 5 is irrational.

Thus, our assumption that 3 + 2 √ 5 is a rational number is wrong.

Hence, 3 + 2 √ 5 is irrational.

Q.5: Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: (i) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 

Note: If the denominator has only factors of 2 and 5 or in the form of 2 m  × 5 n  then it has a terminating decimal expansion. If the denominator has factors other than 2 and 5 then it has a non-terminating repeating decimal expansion.

(i) 13/3125

Factoring the denominator, we get,

3125 = 5 × 5 × 5 × 5 × 5 = 5 5

= 2 0 × 5 5

Since the denominator is of the form 2 m  × 5 n  then, 13/3125 has a terminating decimal expansion.

8 = 2× 2 × 2 = 2 3

= = 2 3 × 5 0

Since the denominator is of the form 2 m  × 5 n  then, 17/8 has a terminating decimal expansion.

(iii) 64/455

455 = 5 × 7 × 13

Since the denominator is not in the form of 2 m  × 5 n , therefore 64/455 has a non-terminating repeating decimal expansion.

(iv) 15/1600

1600 = 2 6  × 5 2

Since the denominator is in the form of 2 m  × 5 n , 15/1600 has a terminating decimal expansion.

Q.6: The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p/q what can you say about the prime factors of q?

(i) 43.123456789

(ii) 0.120120012000120000. . .

(i) 43.123456789 Since it has a terminating decimal expansion, it is a rational number in the form of p/q and q has factors of 2 and 5 only.

(ii) 0.120120012000120000. . . Since it has a non-terminating and non-repeating decimal expansion, it is an irrational number.

Q.7: Check whether 6 n can end with the digit 0 for any natural number n.

If the number 6n ends with the digit zero (0), then it should be divisible by 5, as we know any number with a unit place as 0 or 5 is divisible by 5.

Prime factorization of 6 n  = (2 × 3) n

Therefore, the prime factorization of 6 n doesn’t contain the prime number 5.

Hence, it is clear that for any natural number n, 6 n is not divisible by 5 and thus it proves that 6 n cannot end with the digit 0 for any natural number n.

Q.8: What is the HCF of the smallest prime number and the smallest composite number?

The smallest prime number = 2

The smallest composite number = 4

Prime factorisation of 2 = 2

Prime factorisation of 4 = 2 × 2

HCF(2, 4) = 2

Therefore, the HCF of the smallest prime number and the smallest composite number is 2.

Q.9: Using Euclid’s Algorithm, find the HCF of 2048 and 960.

2048 > 960

Using Euclid’s division algorithm,

2048 = 960 × 2 + 128

960 = 128 × 7 + 64

128 = 64 × 2 + 0

Therefore, the HCF of 2048 and 960 is 64.

Q.10: Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers.

Prime factorisation of 404 = 2 × 2 × 101

Prime factorisation of 96 = 2 × 2 × 2 × 2 × 2 × 3 = 2 5  × 3

HCF = 2 × 2 = 4

LCM = 2 5  × 3 × 101 = 9696

HCF × LCM = 4 × 9696 = 38784

Product of the given two numbers = 404 × 96 = 38784

Hence, verified that LCM × HCF = Product of the given two numbers.

Video Lesson on Numbers

Extra questions for Class 10 Maths Chapter 1

Q.1: Find three rational numbers lying between 0 and 0.1. Find twenty rational numbers between 0 and 0.1. Give a method to determine any number of rational numbers between 0 and 0.1.

Q.2: Which of the following rational numbers have the terminating decimal representation?

(v) 133/125

Q.3: Write the following rational numbers in decimal form:

(v) 327/500

(viii) 11/17

Q.4: If a is a positive rational number and n is a positive integer greater than 1, prove that a n is a rational number.

Q.5: Show that  3 √6 and  3 √3 are not rational numbers.

Q.6: Show that 2 + √2 is not a rational number.

Q.7: Give an example to show that the product of a rational number and an irrational number may be a rational number.

Q.8: Prove that √3 – √2 and √3 + √5 are irrational.

Q.9: Express 7/64, 12/125 and 451/13 in decimal form.

Q.10: Find two irrational numbers lying between √2 and √3.

Q.11: Mention whether the following numbers are rational or irrational:

(i) (√2 + 2)

(ii) (2 – √2) x (2 + √2)

(iii) (√2 + √3) 2

Q.12: Given that √2 is irrational, prove that (5 + 3√2) is an irrational number.

Q.13: Write the smallest number which is divisible by both 360 and 657.

Q.14: Show that the square of any positive integer cannot be of the form (5q + 2) or (5q + 3) for any integer q.

Q.15: Prove that one of every three consecutive positive integers is divisible by 3.

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NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12

Important Questions for Class 10 Maths Chapter 1 Real Numbers

September 19, 2021 by Sastry CBSE

Real Numbers Class 10 Important Questions Very Short Answer (1 Mark)

Question 2. Write the decimal form of \(\frac { 129 }{ { 2 }^{ 7 }{ 5 }^{ 7 }{ 7 }^{ 5 } }\) Solution: Non-terminating non-repeating.

Question 3. Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11, and 15 respectively. Solution: Algorithm 398 – 7 = 391, 436 – 11 = 425, 542 – 15 = 527 HCF of 391, 425, 527 = 17

You can practice more Grade 8 Real Numbers Common Core Questions and Answers to score good marks in the exams.

Question 4. Express 98 as a product of its primes. Solution: 2 × 7 2

Question 5. If the HCF of 408 and 1032 is expressible in the form 1032 × 2 + 408 × p, then find the value of p. Solution: HCF of 408 and 1032 is 24. 1032 × 2 + 408 × (p) = 24 408p = 24 – 2064 p = -5

Real Numbers Class 10 Important Questions Short Answer-I (2 Marks)

Question 6. HCF and LCM of two numbers is 9 and 459 respectively. If one of the numbers is 27, find the other number. (2012) Solution: We know, 1st number × 2nd number = HCF × LCM ⇒ 27 × 2nd number = 9 × 459 ⇒ 2nd number = \(\frac { 9\times 459 }{ 27 }\) = 153

Question 7. Find HCF and LCM of 13 and 17 by prime factorisation method. (2013) Solution: 13 = 1 × 13; 17 = 1 × 17 HCF = 1 and LCM = 13 × 17 = 221

Question 8. Find LCM of numbers whose prime factorisation are expressible as 3 × 5 2 and 3 2 × 7 2 . (2014) Solution: LCM (3 × 5 2 , 3 2 × 7 2 ) = 3 2 × 5 2 × 7 2 = 9 × 25 × 49 = 11025

Question 11. Find the largest number which divides 70 and 125 leaving remainder 5 and 8 respectively. (2015) Solution: It is given that on dividing 70 by the required number, there is a remainder 5. This means that 70 – 5 = 65 is exactly divisible by the required number. Similarly, 125 – 8 = 117 is also exactly divisible by the required number. 65 = 5 × 13 117 = 3 2 × 13 HCF = 13 Required number = 13

Question 12. Find the prime factorisation of the denominator of rational number expressed as \(6.\bar { 12 }\) in simplest form. (2014) Solution: Let x = \(6.\bar { 12 }\) …(i) 100x = 612.\(\bar { 12 }\) …(ii) …[Multiplying both sides by 100] Subtracting (i) from (ii), 99x = 606 x = \(\frac { 606 }{ 99 }\) = \(\frac { 202 }{ 33 }\) Denominator = 33 Prime factorisation = 3 × 11

Question 15. Show that 3√7 is an irrational number. (2016) Solution: Let us assume, to the contrary, that 3√7 is rational. That is, we can find coprime a and b (b ≠ 0) such that 3√7 = \(\frac { a }{ b }\) Rearranging, we get √7 = \(\frac { a }{ 3b }\) Since 3, a and b are integers, \(\frac { a }{ 3b }\) is rational, and so √7 is rational. But this contradicts the fact that √7 is irrational. So, we conclude that 3√7 is irrational.

Question 16. Explain why (17 × 5 × 11 × 3 × 2 + 2 × 11) is a composite number? (2015) Solution: 17 × 5 × 11 × 3 × 2 + 2 × 11 …(i) = 2 × 11 × (17 × 5 × 3 + 1) = 2 × 11 × (255 + 1) = 2 × 11 × 256 Number (i) is divisible by 2, 11 and 256, it has more than 2 prime factors. Therefore (17 × 5 × 11 × 3 × 2 + 2 × 11) is a composite number.

Question 17. Check whether 4n can end with the digit 0 for any natural number n. (2015) Solution: 4 n = (2 2 ) n = 2 2n The only prime in the factorization of 4 n is 2. There is no other prime in the factorization of 4 n = 2 2n (By uniqueness of the Fundamental Theorem of Arithmetic). 5 does not occur in the prime factorization of 4 n for any n. Therefore, 4 n does not end with the digit zero for any natural number n.

Question 18. Can two numbers have 15 as their HCF and 175 as their LCM? Give reasons. (2017 OD) Solution: No, LCM = Product of the highest power of each factor involved in the numbers. HCF = Product of the smallest power of each common factor. We can conclude that LCM is always a multiple of HCF, i.e., LCM = k × HCF We are given that, LCM = 175 and HCF = 15 175 = k × 15 ⇒ 11.67 = k But in this case, LCM ≠ k × HCF Therefore, two numbers cannot have LCM as 175 and HCF as 15.

Real Numbers Class 10 Important Questions Short Answer-II (3 Marks)

Question 19. Prove that √5 is irrational and hence show that 3 + √5 is also irrational. (2012) Solution: Let us assume, to the contrary, that √5 is rational. So, we can find integers p and q (q ≠ 0), such that √5 = \(\frac { p }{ q }\), where p and q are coprime. Squaring both sides, we get 5 = \(\frac { { p }^{ 2 } }{ { q }^{ 2 } }\) ⇒ 5q 2 = p 2 …(i) ⇒ 5 divides p 2 5 divides p So, let p = 5r Putting the value of p in (i), we get 5q 2 = (5r) 2 ⇒ 5q 2 = 25r 2 ⇒ q 2 = 5r 2 ⇒ 5 divides q 2 5 divides q So, p and q have atleast 5 as a common factor. But this contradicts the fact that p and q have no common factor. So, our assumption is wrong, is irrational. √5 is irrational, 3 is a rational number. So, we conclude that 3 + √5 is irrational.

Question 21. Three bells toll at intervals of 9, 12, 15 minutes respectively. If they start tolling together, after what time will they next toll together? (2013) Solution: 9 = 3 2 , 12 = 2 2 × 3, 15 = 3 × 5 LCM = 2 2 × 3 2 × 5 = 4 × 9 × 5 = 180 minutes or 3 hours They will next toll together after 3 hours.

Question 23. The length, breadth, and height of a room are 8 m 50 cm, 6 m 25 cm and 4 m 75 cm respectively. Find the length of the longest rod that can measure the dimensions of the room exactly. (2015) Solution: To find the length of the longest rod that can measure the dimensions of the room exactly, we have to find HCF. L, Length = 8 m 50 cm = 850 cm = 2 1 × 5 2 × 17 B, Breadth = 6 m 25 cm = 625 cm = 5 4 H, Height = 4 m 75 cm = 475 cm = 5 2 × 19 HCF of L, B and H is 5 2 = 25 cm Length of the longest rod = 25 cm

Question 26. By using Euclid’s algorithm, find the largest number which divides 650 and 1170. (2017 OD) Solution: Given numbers are 650 and 1170. 1170 > 650 1170 = 650 × 1 + 520 650 = 520 × 1 + 130 520 = 130 × 4 + 0 HCF = 130 The required largest number is 130.

Real Numbers Class 10 Important Questions Long Answer (4 Marks)

Question 31. Dudhnath has two vessels containing 720 ml and 405 ml of milk respectively. Milk from these containers is poured into glasses of equal capacity to their brim. Find the minimum number of glasses that can be filled. (2014) Solution: 1st vessel = 720 ml; 2nd vessel = 405 ml We find the HCF of 720 and 405 to find the maximum quantity of milk to be filled in one glass. 405 = 3 4 × 5 720 = 2 4 × 3 2 × 5 HCF = 3 2 × 5 = 45 ml = Capacity of glass No. of glasses filled from 1st vessel = \(\frac { 720 }{ 45 }\) = 16 No. of glasses filled from 2nd vessel = \(\frac { 405 }{ 45 }\) = 9 Total number of glasses = 25

Question 32. Amita, Sneha, and Raghav start preparing cards for all persons of an old age home. In order to complete one card, they take 10, 16 and 20 minutes respectively. If all of them started together, after what time will they start preparing a new card together? (2013) Solution: To find the earliest (least) time, they will start preparing a new card together, we find the LCM of 10, 16 and 20. 10 = 2 × 5 16 = 2 4 20 = 2 2 × 5 LCM = 2 4 × 5 = 16 × 5 = 80 minutes They will start preparing a new card together after 80 minutes.

Question 34. If two positive integers x and y are expressible in terms of primes as x = p 2 q 3 and y = p 3 q, what can you say about their LCM and HCF. Is LCM a multiple of HCF? Explain. (2014) Solution: x = p 2 q 3 and y = p 3 q LCM = p 3 q 3 HCF = p 2 q …..(i) Now, LCM = p 3 q 3 ⇒ LCM = pq 2 (p 2 q) ⇒ LCM = pq 2 (HCF) Yes, LCM is a multiple of HCF. Explanation: Let a = 12 = 2 2 × 3 b = 18 = 2 × 3 2 HCF = 2 × 3 = 6 …(ii) LCM = 2 2 × 3 2 = 36 LCM = 6 × 6 LCM = 6 (HCF) …[From (ii)] Here LCM is 6 times HCF.

Question 35. Show that one and only one out of n, (n + 1) and (n + 2) is divisible by 3, where n is any positive integer. (2015) Solution: Let n, n + 1, n + 2 be three consecutive positive integers. We know that n is of the form 3q, 3q + 1, or 3q + 2. Case I. When n = 3q, In this case, n is divisible by 3, but n + 1 and n + 2 are not divisible by 3. Case II. When n = 3q + 1, In this case n + 2 = (3q + 1) + 2 = 3q + 3 = 3(q + 1 ), (n + 2) is divisible by 3, but n and n + 1 are not divisible by 3. Case III. When n = 3q + 2, in this case, n + 1 = (3q + 2) + 1 = 3q + 3 = 3 (q + 1 ), (n + 1) is divisible by 3, but n and n + 2 are not divisible by 3. Hence, one and only one out of n, n + 1 and n + 2 is divisible by 3.

Question 36. Find the HCF and LCM of 306 and 657 and verify that LCM × HCF = Product of the two numbers. (2016 D) Solution: 306 = 2 × 3 2 × 17 657 = 32 × 73 HCF = 3 2 = 9 LCM = 2 × 3 2 × 17 × 73 = 22338 L.H.S. = LCM × HCF = 22338 × 9 = 201042 R.H.S. = Product of two numbers = 306 × 657 = 201042 L.H.S. = R.H.S.

Question 37. Show that any positive odd integer is of the form 41 + 1 or 4q + 3 where q is a positive integer. (2016 OD) Solution: Let a be a positive odd integer By Euclid’s Division algorithm: a = 4q + r …[where q, r are positive integers and 0 ≤ r < 4] a = 4q or 4q + 1 or 4q + 2 or 4q + 3 But 4q and 4q + 2 are both even a is of the form 4q + 1 or 4q + 3.

Important Questions for Class 10 Maths

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Real Numbers

Real number is any number that can be found in the real world. We find numbers everywhere around us. Natural numbers are used for counting objects, rational numbers are used for representing fractions, irrational numbers are used for calculating the square root of a number, integers for measuring temperature, and so on. These different types of numbers make a collection of real numbers. In this lesson, let us learn all about what are real numbers, the subsets of real numbers along with real numbers examples.

What are Real Numbers?

Any number that we can think of, except complex numbers, is a real number. For example, 3, 0, 1.5, 3/2, √5, and so on are real numbers.

Definition of Real Numbers

Real numbers include rational numbers like positive and negative integers, fractions, and irrational numbers. Now, which numbers are not real numbers? The numbers that are neither rational nor irrational are non-real numbers, like, √-1, 2 + 3i, and -i. These numbers include the set of complex numbers , C.

Set of Real Numbers

The set of real numbers, which is denoted by R, is the union of the set of rational numbers (Q) and the set of irrational numbers ( \(\overline{Q}\)). So, we can write the set of real numbers as, R = Q ∪ \(\overline{Q}\). This indicates that real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Observe the following table to understand this better. The table shows the sets of numbers that come under real numbers.

List of Real Numbers

The list of real numbers is endless because it includes all kinds of numbers like whole, natural, integers, rational, and irrational numbers. Since it includes integers it has negative numbers too. So, there is no specific number from which the list of real numbers starts or ends. It goes to infinity towards both sides of the number line.

Types of Real Numbers

We know that real numbers include rational numbers and irrational numbers. Thus, there does not exist any real number that is neither rational nor irrational. It simply means that if we pick up any number from R, it is either rational or irrational.

• Rational Numbers

Any number which can be defined in the form of a fraction p/q is called a rational number. The numerator in the fraction is represented as 'p' and the denominator as 'q', where 'q' is not equal to zero. A rational number can be a natural number, a whole number, a decimal, or an integer. For example, 1/2, -2/3, 0.5, 0.333 are rational numbers.

• Irrational Numbers

Irrational numbers are the set of real numbers that cannot be expressed in the form of a fraction p/q where 'p' and 'q' are integers and the denominator 'q' is not equal to zero (q≠0.). For example, π (pi) is an irrational number. π = 3.14159265...In this case, the decimal value never ends at any point. Therefore, numbers like √2, -√7, and so on are irrational numbers.

Symbol of Real Numbers

Real numbers are represented by the symbol R . Here is a list of the symbols of the other types of numbers.

• N - Natural numbers
• W - Whole numbers
• Z - Integers
• Q - Rational numbers
• \(\overline{Q}\) - Irrational numbers

Subsets of Real Numbers

All numbers except complex numbers are real numbers. Therefore, real numbers have the following five subsets:

• Natural numbers: All positive counting numbers make the set of natural numbers, N = {1, 2, 3, ...}
• Whole numbers: The set of natural numbers along with 0 represents the set of whole numbers. W = {0, 1, 2, 3, ..}
• Integers: All positive counting numbers, negative numbers, and zero make up the set of integers. Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Rational numbers: Numbers that can be written in the form of a fraction p/q, where 'p' and 'q' are integers and 'q' is not equal to zero are rational numbers. Q = {-3, 0, -6, 5/6, 3.23}
• Irrational numbers: The numbers that are square roots of positive rational numbers, cube roots of rational numbers, etc., such as √2, come under the set of irrational numbers. ( \(\overline{Q}\)) = {√2, -√6}

Real Numbers Chart

Among the sets given above, the sets N, W, and Z are the subsets of Q. The following figure shows the real numbers chart that explains the relationship between all the numbers mentioned above.

Properties of Real Numbers

• Closure Property: The closure property states that the sum and product of two real numbers is always a real number. The closure property of R is stated as follows: If a, b ∈ R, a + b ∈ R and ab ∈ R

Associative Property: The sum or product of any three real numbers remains the same even when the grouping of numbers is changed. The associative property of R is stated as follows: If a,b,c ∈ R, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c

Commutative Property: The sum and the product of two real numbers remain the same even after interchanging the order of the numbers. The commutative property of R is stated as follows: If a, b ∈ R, a + b = b + a and a × b = b × a

• Distributive Property: Real numbers satisfy the distributive property. The distributive property of multiplication over addition is, a × (b + c) = (a × b) + (a × c) and the distributive property of multiplication over subtraction is a × (b - c) = (a × b) - (a × c)

Real Numbers on a Number Line

A number line helps us to display numbers by representing them by a unique point on the line. Every point on the number line shows a unique real number. Note the following steps to represent real numbers on a number line:

• Step 1: Draw a horizontal line with arrows on both ends and mark the number 0 in the middle. The number 0 is called the origin.
• Step 2: Mark an equal length on both sides of the origin and label it with a definite scale.
• Step 3: It should be noted that the positive numbers lie on the right side of the origin and the negative numbers lie on the left side of the origin.

Observe the numbers highlighted on the number line. It shows real numbers like -5/2, 0, 3/2, and 2.

Difference Between Real Numbers and Integers

The main difference between real numbers and integers is that real numbers include integers. In other words, integers come under the category of real numbers. Let us understand the difference between real numbers and integers with the help of the following table.

Important Tips on Real Numbers

• Every irrational number is a real number.
• Every rational number is a real number.
• All numbers except complex numbers are real numbers.
• All integers are real numbers.

☛ Related Articles

• Prime Numbers
• Composite Numbers
• Odd Numbers
• Counting Numbers
• Cardinal Numbers
• Even and Odd Numbers
• Sum of Even Numbers
• Even Numbers 1 to 100
• Even Numbers 1 to 1000
• Odd and Even Numbers Worksheets
• Natural Numbers
• Decimal Representation of Irrational numbers
• Complex Conjugate

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Real Numbers Examples

Example 1: Identify the real numbers among the following: √6, -3, 3.15, -1/2, √-5.

Among the given numbers, √-5 is a complex number. Hence, it cannot be a real number. The other numbers are either rational or irrational. Thus, they are real numbers. Therefore, the real numbers from the list are √6, -3, 3.15, and -1/2

Example 2: Fill in the blanks with respect to real numbers:

a.) Real numbers are denoted by _.

b.) Positive real numbers start from _

a.) Real numbers are denoted by the letter R.

b.) Positive real numbers start from 1 because positive numbers mean numbers that are greater than 1. Otherwise, there is no specific number from which the list of real numbers starts or ends. It goes to infinity towards both sides of the number line.

Example 3: State true or false with respect to real numbers.

a.) Every irrational number is a real number.

b.) Every rational number is a real number.

c.) Every real number is a complex number.

d.) Every real number is an irrational number.

e.) Every natural number is a real number.

f.) Every real number is a rational number.

a.) True, every irrational number is a real number.

b.) True, every rational number is a real number.

c.) False, complex numbers are not real numbers.

d.) False, every real number is not an irrational number, it can be a natural number, a whole number or an integer.

e.) True, every natural number is a real number.

f.) False, every real number may not necessarily be a rational number because real numbers include irrational numbers as well.

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Practice Questions on Real Numbers

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FAQs on Real Numbers

What are real numbers in math.

Real numbers include rational numbers like positive and negative integers , fractions, and irrational numbers . In other words, any number that we can think of, except complex numbers , is a real number. For example, 3, 0, 1.5, 3/2, √5, and so on are real numbers.

What are the Properties of Real Numbers?

The set of real numbers satisfies the closure property, the associative property, the commutative property, and the distributive property.

Closure Property: The sum and product of two real numbers is always a real number. The closure property of R is stated as follows: If a, b ∈ R, a + b ∈ R and ab ∈ R

• Distributive Property: The distributive property of multiplication over addition is a × (b + c) = (a × b) + (a × c) and the distributive property of multiplication over subtraction is a × (b - c) = (a × b) - (a × c)

What are the Subsets of Real Numbers?

Real numbers have the following five subsets:

• Natural numbers: N = {1, 2, 3, ...}
• Whole numbers: W = {0, 1, 2, 3, ..}
• Integers: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Rational numbers: Q = {-3, 0, -6, 5/6, 3.23}
• Irrational numbers: ( \(\overline{Q}\)) = {√2, -√6}

What are Non Real Numbers?

Complex numbers, like √-1, are not real numbers. In other words, the numbers that are neither rational nor irrational, are non-real numbers.

How to Classify Real Numbers?

Real numbers can be classified into two types, rational numbers and irrational numbers. A rational number includes positive and negative integers, fractions , like, -2, 0, -4, 2/6, 4.51, whereas, irrational numbers include the square roots of rational numbers, cube roots of rational numbers, etc., such as √2, -√8

How to Represent Real Numbers on Number Line?

Real numbers can be represented on a number line by following the steps given below:

• Draw a horizontal line with arrows on both ends and mark the number 0 in the middle. The number 0 is called the origin.
• Mark an equal length on both sides of the origin and label it with a definite scale.
• Remember that the positive numbers lie on the right side of the origin and the negative numbers lie on the left side of the origin.

Is the Square Root of a Negative Number a Real Number?

No, the square root of a negative number is not a real number. For example, √-2 is not a real number. However, if the number inside the √ symbol is positive, then it will be a real number.

Is 0 a Real Number?

Yes, 0 is a real number because it belongs to the set of whole numbers and the set of whole numbers is a subset of real numbers.

Is 9 a Real Number?

Yes, 9 is a real number because it belongs to the set of natural numbers that comes under real numbers.

What is the Difference Between Real Numbers, Integers, Rational Numbers, and Irrational Numbers?

The main difference between real numbers and the other given numbers is that real numbers include rational numbers, irrational numbers, and integers. For example, 2, -3/4, 0.5, √2 are real numbers.

• Integers include only positive numbers, negative numbers, and zero. For example, -7,-6, 0, 3, 1 are integers.
• Rational numbers are those numbers that can be written in the form of a fraction p/q, where 'p' and 'q' are integers and 'q' is not equal to zero. For example, -3, 0, -6, 5/6, 3.23 are rational numbers.
• Irrational numbers are those numbers that are square roots of positive rational numbers, cube roots of rational numbers, etc., such as √2, - √5, and so on.

Where do Real Numbers Start from?

There is no specific number from where real numbers start or end because real numbers include all kinds of numbers like whole, natural, integers, rational, and irrational numbers except complex numbers. Since it includes integers it has negative numbers as well. So, the list of real numbers goes to infinity towards both sides of the number line.

Real Number System Test Quiz!

. Dive into the universe of numbers with the Real Number System Test Quiz! This comprehensive assessment is designed to probe your understanding of real numbers, spanning rational and irrational realms. Explore the intricacies of integers, fractions, and decimals, distinguishing between elements that fall within the real number spectrum. Tackle challenging questions that navigate through concepts like absolute values, ordering, and arithmetic operations on real numbers. Whether you're a math whiz or a budding enthusiast, this quiz is a litmus test for your comprehension of the fundamental building blocks of mathematics. Unleash your numerical prowess and conquer the Real Number Read more System Test Quiz!

18 is a whole number.

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− 3/2 is an integer.

2.434434443… is a rational number., 6.57 is an integer., 5. 7777 is rational. , all fractions are rational numbers. , all integers are whole numbers. , all irrational numbers are real numbers., all negative whole numbers are integers., which of these sets of numbers contain all rational numbers .

π ,1,2, -13

-3.0541... , 99, 0.14363

-6, − 225, 4,7,8

21, 0.75, 0, √2

Which of these sets of numbers contains all rational numbers?

π,1,2, -13

21, 0.75, 0

Given the following set of numbers, circle each irrational number (there may be more than one).

−5 is a rational number. , 0 is an integer. , square root of 16 is a natural number., −3. 25 is an integer., square root of 8 is rational., square root of 7 is a real number., what is the best classification for -4.

integer, rational number, real number

irrational number, real number

Whole number, integer, real number

Rational number, real number

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The resources on this page will hopefully help you teach AO2 and AO3 of the new GCSE specification - problem solving and reasoning.

This brief lesson is designed to lead students into thinking about how to solve mathematical problems. It features ideas of strategies to use, clear steps to follow and plenty of opportunities for discussion.

The PixiMaths problem solving booklets are aimed at "crossover" marks (questions that will be on both higher and foundation) so will be accessed by most students. The booklets are collated Edexcel exam questions; you may well recognise them from elsewhere. Each booklet has 70 marks worth of questions and will probably last two lessons, including time to go through answers with your students. There is one for each area of the new GCSE specification and they are designed to complement the PixiMaths year 11 SOL.

These problem solving starter packs are great to support students with problem solving skills. I've used them this year for two out of four lessons each week, then used Numeracy Ninjas as starters for the other two lessons.  When I first introduced the booklets, I encouraged my students to use scaffolds like those mentioned here , then gradually weaned them off the scaffolds. I give students some time to work independently, then time to discuss with their peers, then we go through it as a class. The levels correspond very roughly to the new GCSE grades.

Some of my favourite websites have plenty of other excellent resources to support you and your students in these assessment objectives.

@TessMaths has written some great stuff for BBC Bitesize.

There are some intersting though-provoking problems at Open Middle.

I'm sure you've seen it before, but if not, check it out now! Nrich is where it's at if your want to provide enrichment and problem solving in your lessons.

MathsBot  by @StudyMaths has everything, and if you scroll to the bottom of the homepage you'll find puzzles and problem solving too.

I may be a little biased because I love Edexcel, but these question packs are really useful.

The UKMT has a mentoring scheme that provides fantastic problem solving resources , all complete with answers.

I have only recently been shown Maths Problem Solving and it is awesome - there are links to problem solving resources for all areas of maths, as well as plenty of general problem solving too. Definitely worth exploring!

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Skeleton Shapes

Skeleton shapes are made with balls of modelling clay and straws.

This shows a cube and a skeleton cube:

How many balls of modelling clay and how many straws does it take to make the cube?

Here are some piles of modelling clay balls and straws:

Look at the shapes below and decide which piles are needed to make a skeleton of each shape.

How do you know which piles go with which shape?

We had lots of correct solutions to this problem and you used many different ways of helping yourselves to reach the answers. Some of you made the shapes from straws and modelling clay, like Rachel, Abigail and Alistair from Histon and Impington Infant School. They sent in a photo of their models:

Others of you decided to make the shapes from Polydron, while some of you drew the shapes carefully. Alice from Tattingstone, and Scarlett and Sam from Cupernham, were among those who opted for drawing, but William, also from Cupernham sent in particularly careful sketches using isometric paper:

Chris and Michael from Moorfield Junior School also sent some excellent drawings and took the problem a step further by looking at the relationship between the number of faces, edges and vertices of different prisms. They enclosed this table:

Chris and Michael say:

There are patterns you can see in the vertices and edges columns. The number of vertices is double the amount of sides on the 2D shape at each end. In the edges column it's three times the amount of sides on the 2D shape.

Perhaps you can spot some more patterns too. Let us know if so.

Thank you also to these people who sent us correct answers too:

Hugo, Emile and Benny from Wesley College Prahran Preparatory School in Melbourne, Australia. Charlotte and Thomas. Al from Dudley. Ryo, Jake and Charlie from Moorfield Junior School. Samuel from Bispham Drive Junior School in Nottingham. David from Tithe Barn Primary School.

Why do this problem?

This problem helps children begin to become familiar with the various properties of common geometric solid shapes, concentrating on edges and vertices. It also helps in promoting discussion and experimentation. Naming the shapes should be a help during discussion and description of what has been done, rather than being an exercise in its own right.

Possible approach

Before doing this problem children should have had plenty of free play building with sets of solid shapes so that they begin to get a feel for their properties. They should also have chance to experiment with building skeleton shapes either with a specific kit or with drinking straws and modelling clay/plasticine.

Key questions

How many edges did you count? What does this tell you about the number of straws we need? How many corners did you count? What does this tell you about the number of balls of modelling clay we need? How many edges meet at this corner?

Possible extension

Possible support.

Start by counting the faces on a cube - a large dice might be useful - and then the edges and finally the vertices. A non-permanent pen could be used to mark a real shape if children are having difficulty keeping track of their counting.