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Standard Form: Understanding the Basics

Why standard form is important, converting to and from standard form, working with standard form in equations, practical applications of standard form, tips and tricks for mastering standard form, what is standard form in math: step-by-step guide.

Welcome to the fascinating world of Standard Form, a critical concept in mathematics that offers a simplified way to represent very large or very small numbers, as well as a specific format for writing linear equations.

Understanding Standard Form is not just an academic exercise—it's a crucial skill with practical applications in science, engineering, finance, and more. Through this article, we aim to demystify Standard Form, making it accessible and understandable.

Before diving into the complexities and applications of Standard Form, it's essential to build a solid foundation by understanding its basic principles and components.

Standard Form can refer to two different concepts in mathematics: a way to write very large or very small numbers using powers of ten, and a specific method to express linear equations. Let’s break down both meanings.

Definition of Standard Form

Standard Form of Numbers : In mathematics, Standard Form is an alternative method for representing numbers that are either excessively vast or small to be expressed in decimal form. It is expressed as \(a \times 10^n\), where \(1 \leq |a| < 10\) and \(n\) is an integer. This method is particularly useful in scientific and engineering fields where dealing with extremely large or small values is common.

Example : The speed of light in a vacuum is approximately 299,792,458 meters per second. In Standard Form, this is written as \(2.99792458 \times 10^8\) m/s.

Standard Form of Linear Equations : In the context of algebra, a linear equation in Standard Form is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) and \(B\) are not both zero. This form is useful for analyzing linear relationships and solving for variables.

Example : \(2x + 3y = 6\) is a linear equation in Standard Form.

Components of Standard Form

For Numbers :

- \(a\) : The coefficient, which must be a value between 1 and 10 (including 1 but not 10). It represents the significant figures of the original number.

- \(10^n\) : The base and exponent component, where \(10\) is the base and \(n\) is the integer exponent. This part dictates the decimal point's movement to convert the number into its original value.

For Linear Equations :

- \(A\) and \(B\) : Coefficients of the variables \(x\) and \(y\), respectively. These coefficients are integers and dictate the slope and intercept of the linear equation.

- \(C\) : The constant term, an integer that represents the point at which the line crosses the y-axis when \(x = 0\).

Understanding and using Standard Form is not just a mathematical exercise—it has practical significance in many areas of life. Let’s explore why mastering this concept is so beneficial.

Simplification of Complex Numbers

Standard Form allows us to simplify the representation of very large or small numbers, making them easier to read, understand, and use in calculations. For instance, comparing the sizes of planets, distances between celestial bodies, or even quantities in molecular biology becomes manageable with Standard Form.

Universality in Science and Engineering

In scientific research and engineering, Standard Form is a universal language. It ensures clarity and precision in measurements, calculations, and sharing data across global teams, facilitating collaboration and innovation.

Financial Applications

In finance, Standard Form can represent large transactions, global currencies, and economic indicators in a concise, manageable way. This simplification aids in analysis, forecasting, and decision-making.

Educational Value

Learning Standard Form equips students with a versatile tool for their mathematical toolkit, enhancing their problem-solving abilities and preparing them for advanced studies in various fields.

By bridishing a clear understanding and appreciation for Standard Form, students and professionals alike can navigate numerical complexities with greater ease and accuracy.

Mastering the conversion of numbers to and from Standard Form not only deepens your understanding of the concept but also enhances your agility in handling mathematical and real-world problems. Here's how:

Converting Large Numbers to Standard Form

To convert a large number to Standard Form, follow these steps:

1. Place the decimal after the first non-zero digit.

2. Count the number of places you moved the decimal to reach its new position. This number becomes \(n\), the exponent of \(10\).

3. Write the number as \(a \times 10^n\), where \(a\) includes all the significant figures of the original number.

Example : Convert 123,400,000 to Standard Form.

- New form: \(1.234\)

- Decimal moves 8 places: \(1.234 \times 10^8\)

Converting Small Numbers to Standard Form

Small numbers are converted similarly, but the exponent on \(10\) will be negative, reflecting the number’s value being less than one.

1. Move the decimal to the right of the first non-zero digit.

2. Count the places moved—this becomes your negative exponent.

3. Represent as \(a \times 10^{-n}\).

Example : Convert 0.00056 to Standard Form.

- New form: \(5.6\)

- Decimal moves 4 places: \(5.6 \times 10^{-4}\)

Converting from Standard Form to Ordinary Numbers

To convert a number from Standard Form back to its original (or ordinary) form, reverse the process:

- If \(n\) is positive, move the decimal point to the right \(n\) times.

- If \(n\) is negative, move the decimal point to the left \(-n\) times.

Example : Convert \(3.2 \times 10^3\) to ordinary form.

- Move decimal 3 places to the right: 3200.

And : Convert \(4.7 \times 10^{-2}\) to ordinary form.

- Move decimal 2 places to the left: 0.047.

A fundamental aspect of algebra is manipulating and solving equations, including those in Standard Form. Understanding how to work with linear equations in Standard Form \(Ax + By = C\) can simplify problem-solving and analysis in various mathematical contexts.

Recognizing Standard Form in Equations

A linear equation is said to be in Standard Form when it follows the \(Ax + By = C\) format, where:

- \(A\), \(B\), and \(C\) are integers,

- \(A\) should be a positive integer,

- \(A\) and \(B\) are not both zero.

Example : The equation \(2x + 3y = 12\) is in Standard Form. Here, \(A = 2\), \(B = 3\), and \(C = 12\).

Solving Equations in Standard Form

Solving Standard Form equations often involves rearranging the equation to solve for one variable (e.g., \(y\)) in terms of the other (\(x\)), which can then be graphed or further manipulated as needed.

1. Isolate one variable : If solving for \(y\), move \(x\) to one side.

\(Ax + By = C \Rightarrow By = -Ax + C\)

2. Divide by the coefficient of the isolate variable : To solve for \(y\), divide the entire equation by \(B\).

\(y = \frac{-A}{B}x + \frac{C}{B}\)

Example Problem : Solve for \(y\) in \(2x + 3y = 12\).

1. Isolate \(y\): \(3y = -2x + 12\)

2. Solve for \(y\): \(y = \frac{-2}{3}x + 4\)

Understanding and applying Standard Form extends beyond the classroom; it has real-world implications that affect how we interpret, interact with, and even shape the world around us.

Scientific and Engineering Applications

In fields like physics, chemistry, astronomy, and engineering, Standard Form is invaluable for dealing with the extremely large or small quantities often involved. For instance, astronomers use Standard Form to denote distances between celestial bodies, while engineers may use it to calculate the minuscule tolerances required in precision machinery.

Economic and Financial Contexts

Standard Form simplifies the representation and comparison of large financial figures, such as the Gross Domestic Product (GDP) of countries or the value of investments. This simplification is crucial for clear communication among stakeholders and policy-making.

Everyday Examples

Even outside of professional environments, Standard Form pops up in consumer technology, such as calculating the storage capacity of devices (expressed in gigabytes, terabytes, etc., which are essentially Standard Forms) or understanding the scale of microscopic or cosmic phenomena through documentaries and educational materials.

Mastering Standard Form, like any mathematical concept, requires practice and a good strategy. Here are some tips and tricks to help you become more proficient:

Focus on the Basics

- Understand the Components : Make sure you have a solid grasp of what constitutes Standard Form, both for numbers and equations. Knowing the role of each component will make conversions and manipulations easier.

- Memorize Key Exponents : Familiarize yourself with the powers of 10 up to at least 10^9 and down to 10^-9. This can speed up the process of converting to and from Standard Form.

Practice Regularly

- Convert Back and Forth : Take ordinary numbers and practice converting them to Standard Form, and vice versa. This will help you get comfortable with moving the decimal point and determining the correct exponent.

- Use Real-World Examples : Apply Standard Form to real-life scenarios, such as calculating distances in the solar system or the sizes of microscopic objects. This not only reinforces learning but also shows the practical value of what you're learning.

Utilize Resources

- Online Tools and Calculators : Use online calculators and conversion tools to check your work. While it's important to understand how to perform conversions manually, these tools can provide quick verification.

- Educational Videos and Websites : Visual and interactive resources can be extremely helpful for grasping conceptual information and seeing examples worked out.

Engage with Peers

- Study Groups : Joining a study group can provide support, new perspectives, and motivation. Explaining concepts to others is also one of the best ways to solidify your understanding.

Standard Form is a cornerstone of mathematical literacy, providing a streamlined method for representing and manipulating very large or very small numbers, as well as a structured approach to solving linear equations.

Its importance cannot be overstated, stretching across academic disciplines and into the realms of science, engineering, finance, and daily life. By understanding and mastering Standard Form, you can unlock new levels of numerical comprehension, enhance your analytical skills, and approach complex problems with confidence.

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Standard Form

Standard form in maths is the method of representing a particular element in the most common manner. From large numbers to small numbers to equations to lines, every element in maths is denoted in a standard form. Let us explore this interesting concept of standard form in various elements of maths such as fractions, equations, algebra, slope along with learning the standard from formula. Solving examples and understanding the basic thumb rule will help in understanding the concept better.

What is Standard Form?

A standard form is a form of writing a given mathematical concept like an equation, number, or an expression in a form that follows certain rules. Representing very large or very small numbers concisely, the standard form is used. For example, 4.5 billion years is written as 4,500,000,000 years. As you can see here, writing a large number like 4.5 billion in its number form is not just ambiguous but also time-consuming and there are chances that we may write a few 0’s less or more while writing in the number form. In this case, writing the number in standard form is very helpful. For example, the standard form of 4,500,000,000 = 4.5 × 10 9 . Not only numbers but the fractions, equations, expressions, polynomials, etc also can be written in the standard form.

Let us study the standard form of each of these in detail.

Standard Form of Number

The standard form of numbers (which is also known as " scientific notation " of numbers)  has different meanings depending on which country you are in. In the United Kingdom and countries using UK conventions, the standard form is another name for scientific notation. Scientific notation is the process of writing a very large or very small number using numbers between 1 to 10 multiplied by the power of 10. For example, 3890 is written as 3.89 × 10 3 . These are numbers that are greater than 1 use positive powers of 10. Numbers less than 1 use the negative power of 10. For example, 0.0451 is written as 4.51 × 10 -2 .

In the United States and countries using US conventions, the standard form is the usual way of writing numbers in decimal notation . Using the same example,

Standard form = 3890, Expanded form = 3000 + 800 + 90, and Written form = Three thousand eight hundred and ninety.

Standard Form of Fraction

In the case of fractions, we need to ensure that in the standard form of fractions, the numerator and denominator must be co-prime numbers . That means they have no common factor other than 1, hence the standard form is also called the simplest form of a fraction . For example, 14/22 and 13/6. The simplest form of 14/22 = 7/22 and 13/6 is already in its simplest form as 13 and 6 are co-prime.

Standard Form of Equations

The standard form of an equation is where zero goes on the right and everything else goes on the left. i.e., it is of the form

• Expression = 0.

This helps in solving the equation in a simple manner. Equations used for polynomials, linear and quadratic have a standard form, let us look at what they are.

Standard Form of Polynomial

The standard form polynomial is written with the exponents in decreasing order to make calculations easier. A polynomial is said to be in its standard form, if it is expressed in such a way that the term with the highest degree is placed first, followed by the term which has the next highest degree, and so on. There are two very simple rules of writing a polynomial in a standard form, they are:

• Write the terms in the descending order of their powers (also called exponents).
• Ensure the polynomial contains no like terms.

Hence, the standard form is a n x n  + a n-1 x n-1  + a n-2 x n-2  + ... + a 1 x 1  + a 0 . For example, the standard form of equation y 2 + 7y 6 - 8y - y 9 is written as -y 9 + 7y 6 + y 2 - 8y.

Note: The thumb rule for writing a polynomial in its standard form is D-U. D stands for Descending and U stands for unlike terms.

Standard Form of a Linear Equation

The standard form of linear equations also known as the general form is a method of writing linear equations. A linear equation can be written in different forms like the standard form, the slope-intercept form, and the point-slope form . The standard form of a linear equation is expressed in two ways, with one variable and with two variables . The standard form of linear equation with one variable is expressed as ax + b = 0 where a and b are constants and the letters x is the variable. The standard form of a linear equation with two variables is expressed as ax + by = c. where a, b, and c are real numbers and the letters x and y are the variables. Look at the image below.

Let’s see how to convert the two lines into the standard form of a linear equation ax + by = c.

Line 1: x + y = 7 i.e. 1x +1y = 7. Here, a = 1, b =1 , c = 7

Line 2: y = 3x i.e. 3x -1y = 0. Here, a = 3, b = -1 , c = 0

Therefore, what we see here is the standard form of equation which is linear i.e. ax + by = c.

Standard Form of Slope of a Line

The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line. Representing a line geometrically, we use the standard form of a linear equation (mentioned above). To determine the slope of a line that is expressed graphically, the equation must be converted to a slope-intercept form . To do this, we must solve the equation for y, and the standard form of a slope is expressed as y = mx + c, where m is the slope of the line. This formula is used when the line is straight.

When there are two points in a plane , the slope can be defined as the ratio of change in the value of y to the change in the value of x. The standard form of slope of a line is expressed as m = (y 2 – y 1 )/(x 2 – x 1 ). The image below represents both the coordinates on a graph.

Standard Form of Quadratic Equations

The standard form of quadratic equation in a variable x is of the form ax 2 + bx + c = 0, where a ≠ 0, and a, b, and c are real numbers . Here, b and c can be either zeros or non-zero numbers and

• 'a' is the coefficient of x 2
• 'b' is the coefficient of x
• 'c' is the constant

Apart from the standard form of a quadratic equation, a quadratic equation can be written in several other forms.

• Vertex Form: a (x - h) 2 + k = 0
• Intercept Form: a (x - p)(x - q) = 0

A parabola is a graph of a quadratic function that refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed-line. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola . The following graphs are two typical parabolas; their x and y-intercepts are marked by green dots, and the vertex of each parabola is marked by a blue dot.

The graph will have one of the two shapes as shown above.

• When a > 0, it represents the first parabola (figure 1) which opens upward (U-shaped).
• When a < 0, we obtain a parabola that opens downward (inverted U-shaped) as shown in figure 2.

When we put y = 0 in the above equation, we get the x-intercept which is also called the roots of the equation . Finding the roots provides us with the turning point of the parabola and the vertical line which is drawn from the turning point is called the axis of symmetry . Look at the image below for reference.

Standard Form Formula

The standard form formula represents the standard form of an equation which is the commonly accepted form of an equation. The formula to represent the standard form formula is based on the degree of the equations.

The standard form of a linear equation is the basic form of an equation. The standard form of a linear equation with two variables and more than two variables is presented below. Here x, y, or x 1 , x 2 , x 3 ,.... represent the variables and a, b, a 1 , a 2 , a 3 , ........ a n are referred to as the coefficients. The numerics placed to the right of the equals to sign are called constants.

ax + by = c

a 1 x 1 +a 2 x 2 + a 3 x 3 + ........ + a n x n = D

The standard form of a quadratic equation is a second-degree equation and has a variable, coefficients, and constant term. Here it is a single variable x of degree 2. The standard form of a quadratic equation is ax 2 + bx + c = 0, where a ≠ 0.

Further, we have standard form formulas for equations of higher degrees. Also in coordinate geometry , we have a standard form for different geometric representations such as a straight line, circle , ellipse , hyperbola , and parabola.

• Straight-line: ax + by = c, where a is a positive integer , and b, and c are integers.
• Circle: (x - h) 2 + (y - k) 2 = (r) 2 , where ( h, k) is the center and r is the radius .
• Ellipse: x 2 /a 2  + y 2 /b 2  = 1
• Hyperbola: (x-x 0 ) 2 /a 2  - (y-y 0 ) 2 /b 2  = 1, where x 0 , x 0 are the center points, a = semi-major axis and b = semi-minor axis.
• Parabola: (x - h) 2 = 4p(y - k)

☛Related Topics

Listed below are a few topics that are related to a standard form.

• Standard Form Calculator
• Standard Form to Vertex Form
• Polynomial in One Variable in Standard Form

Standard Form Examples

Example 1: Liza is trying to find out which of the following equations represent the given graph. As there are no values of the coordinates given, she is not able to decide. How can we use the standard form concept to solve her problem?

1. x 4 + 7x 2 - 5x = 4

2. 4x + 5y = 0

3. y 2 + 7y 6 - 9y = y 3

4. 3z 4 + 7z 5 = -12z

The graph given here represents a line . Now, a line represents a linear equation in two variables which has a degree of 1. Out of the four equations given above, only option (b) is linear.

Answer:  Therefore, the graph represents the equation, 4x + 5y = 0.

Example 2: Anna showed her class notes on polynomials to her teacher. The teacher returned her notes with a remark "Write the polynomial x 2 -10x + 16- x 2 + x 5 - 3x 4 + 3x 2 in standard form." What is the correct format Anna should have used?

To write a polynomial in standard form, two rules must be taken care of.

1. Write the terms in descending order of their powers.

2. All terms must be unlike.

Let us first arrange in descending order .

x 2 - 10x + 16- x 2 + x 5 - 3x 4 + 3x 2 = x 5 - 3x 4 + x 2 - x 2 + 3x 2 - 10x +16

After adding like terms , we get

x 5 - 3x 4 +3x 2 - 10x + 16.

Answer:  Therefore, the standard form is x 5 - 3x 4 +3x 2 - 10x + 16.

Example 3: Convert the following quadratic equation into standard form: (1/x) + x = 1.

(1/x) + x = 1

Multiplying with x on both sides,

1 + x 2 = x

Shifting R.H.S. terms to L.H.S.,

x 2 - x + 1 = 0

Answer:  Therefore, Standard form of the given quadratic equation is x 2 - x + 1 = 0.

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Practice Questions on Standard Form

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FAQs on Standard Form

What is the definition of standard form in math.

Standard form in math is the method of representing a particular element (numbers, fractions, equations, etc) in the most common way. Very large numbers or very small numbers are expressed in the standard form. Mathematical elements such as equations are expressed in a standard form to better solve the problem. In other words, a standard form is a form of writing a given mathematical concept like an equation, number, or expression in a form that follows certain rules.

How Do We Write Standard Form in Maths?

The process of writing a given mathematical concept like an equation, number, or expression in certain rules is called the standard form. Depending upon which mathematical concept we are dealing with, the standard form formula will vary.

How Do You Write Standard Form of Number?

Standard form in math of numbers is written differently depending on the country. In the UK, numbers that are greater than 1 use positive powers of 10, and numbers less than 1 use the negative power of 10. For example, 3670000 is written as 3.67 × 10 6 and 0.0763 is written as 7.63 × 10 -2 . This is commonly known as scientific notation of numbers.

What is Standard Form Formula?

The standard form formula refers to the formula presenting the general representation for different types of notation. For example, the standard form of

• a linear equation is ax + by = c.
• a quadratic equation is ax 2  + bx + c = 0
• a cubic equation is ax 3  + bx 2  + cx + d = 0

What is the Standard Form Formula for Parabola?

The standard form formula of the equation of the parabola is this: (y - k) 2 = 4p(x - h), where p≠ 0 only in case a parabola has a horizontal axis.

• The vertex of this parabola is at (h, k).
• The focus is at (h + p, k).

What is the Standard Form for Slope Formula?

The standard form of the  slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line whereas the standard form of a linear equation is Ax + By = C and the slope in this form is -A/B.

How to Use Standard Form Formula?

We can use the standard form formula depending on the equation, if it's linear, quadratic, etc. Just rewrite the given formulas in the standard form.

• Standard form of linear equation : ax + by = c
• Standard form of a quadratic equation is a second degree equation: ax 2 + bx + c = 0

What is the Standard Form of Equation?

In the standard form of an equation, 0 is usually present on the right side, whereas the rest of the expressions are on the left. Also, the terms are arranged in the descending order of their exponents. To convert an equation into stadard form, just apply arithmetic operations on both sides to turn the right side to be 0.

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Standard Form

Converting to and from Standard Form

Here we will learn how to convert to and from standard form , including how to adjust numbers to write them in standard form notation.

There are also converting to and from standard form worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is converting to and from standard form?

Converting to and from standard form is where we convert an ordinary number to a number written in standard form or scientific notation.

Standard form is a way of writing very large or very small numbers by comparing the powers of ten. Numbers in standard form are written in this format:

Where a is a number 1\leq{a}\lt10 and n is an integer.

Converting between ordinary numbers and numbers in standard form can help us to compare numbers and interpret answers given in standard form on a calculator. To do this we need to understand the place value of a number.

E.g. Let’s look at the number 350000 and place the digits in a place value table:

So 350000 written in standard form is:

How to convert ordinary numbers to standard form

In order to convert ordinary numbers to standard form:

• Identify the non-zero digits and write these as a decimal number which is \pmb{ 1\leq{x}\lt10} .
• In order to maintain the place value of the number, this decimal number needs to be multiplied by a power of ten.

Write the power of ten as an exponent.

Explain how to convert ordinary numbers to standard form

Standard form calculator worksheet

Get your free standard form calculator worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Converting ordinary numbers to standard form examples

Example 1: writing numbers in standard form with positive powers.

Write this number in standard form:

The non-zero digits need to be written in decimal notation.

The number needs to lie between 1\leq{x}\lt10

So the number will begin as 4.8… .

2 You now need to maintain the value of the number by multiplying that decimal by a power of ten. 

3 Write that power of ten as an exponent.

4 Write your number in standard form.

Example 2: writing numbers in standard form with positive powers

The number needs to lie between 1\leq{x}\lt10

So the number will begin as 5.42… .

You now need to maintain the value of the number by multiplying that decimal by a power of ten. 

Write that power of ten as an exponent.

Write your number in standard form.

Example 3: converting a small number to standard form

Write 0.00081 in standard form.

Identify the non-zero digits and write these as a decimal number which is \pmb{1\leq{x}\lt10}.

This number will begin with 8.1...

Identify what power of ten the decimal needs to be multiplied by in order to preserve place value .

8.1 \div 10000 = 0.00081

Example 4: converting a small number to standard form

Write 0.00718 in standard form.

Identify the non-zero digits and write these as a decimal number which is \pmb{{1\leq{x}\lt10}.}

The number will begin with 7.18…

7.8 \div 1000 = 0.0078

How to convert standard form to ordinary numbers

In order to convert from standard form to ordinary numbers:

• Convert the power of ten to an ordinary number
• Multiply the decimal number by this power of ten
• Write your number as an ordinary number

Explain how to convert standard form to ordinary numbers

Converting standard form to ordinary numbers examples

Example 5: converting standard form to an ordinary number.

Write 6.2\times10^4 as an ordinary number.

Write the exponent as a power of ten.

Multiply the decimal number by that power of ten.

Write your answer as an ordinary number.

Example 6: converting standard form to an ordinary number

Write 1.9\times10^{-3} as an ordinary number.

How to adjust numbers to standard form

Sometimes we might have a number that looks like it is in standard form however the decimal number is not between 1 and 10 , E.g. 36103 or 0.2104 . In this case we need to adjust the number.

In order to adjust numbers to standard form:

• Identify what power of ten the decimal number needs to be multiplied by so that the value is \pmb{1\leq{x}\lt10}.
• Apply the inverse of this to the power of ten.

Adjusting numbers to standard form examples

Example 7: adjusting number in standard form.

Write 48\times10^5 in standard form.

Identify what the first number needs to be multiplied or divided by so that it lies between \pmb{1\leq{x}\lt10}.

48 needs to be divided by 10 so 48 becomes 4.8 .

Apply the inverse operation to the power of ten.

10^5 needs to be multiplied by 10 which adds one to its power, so it becomes 10^6 .

Write your number in standard form. 

Example 8: adjusting numbers to standard form

Write 0.68\times10^{4} in standard form.

0.68 needs to be multiplied by 10 so it becomes 6.8 .

10^4 needs to be divided by 10 which subtracts one from its power, so it becomes 10^3 .

Example 9: adjusting numbers to standard form

Write 290\times10^{-4} in standard form.

290 needs to be divided by 100 so it becomes 2.9 .

10^{-4} needs to be multiplied by 100 which adds two to its power, so it becomes 10^{-2} .

Common misconceptions

• Writing a number with the incorrect power for a large or small number

This error is often made by counting the zeros following the first non zero digit for large numbers or zeros after the decimal point for small numbers, then writing this as the power, rather than considering the place value of the given number.

• Identifying incorrect place value with small numbers

In a number such as 0.000682 , selecting the ‘2’ to determine the exponent rather than the ‘6’ which has a higher place value. In standard form, this number would be 6.82 × 10^{-4} .

• Errors with negative numbers

When checking the standard form of a number, incorrectly adjusting the negative powers due to not applying negative numbers rules correctly. E.g. With small numbers, adding one to the power of 10^{-5} will result in 10^{-4} not 10^{-6} .

Related lessons

Converting to and from standard form is part of our series of lessons to support revision on standard form. You may find it helpful to start with the main standard form lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Standard form
• Multiplying and dividing in standard form
• Adding and subtracting in standard form

Practice standard form calculator questions

1. Write 270000 in standard form

The number between 1 and 10 here is 2.7.

2. Write 0.00079 in standard form

The number between 1 and 10 here is 7.9.

3. Write 6.1 \times 10^{4} as an ordinary number

4. Write 3.8 \times 10^{-5} as an ordinary number

5. Write 84\times10^{2} in standard form

This number is not in standard form as 84 is not between 1 and 10.

We need to divide 84 by 10 and, to compensate, multiply 10^2 by 10 , increasing the power by 1.

This gives us

6. Write 0.92\times10^{-5} in standard form

This number is not in standard form because 0.92 is not between 1 and 10.

We need to multiply 0.92 by 10 and, to compensate, divide 10^{-5} by 10 , decreasing the power by 1.

Standard form calculator GCSE questions

  (a)  Write 8.23\times10^{-6} , as an ordinary number.

(b)  Write the number 0.00702 in standard form.

(a)  0.00000823

(b)  7.02\times10^{-3}

(a)  The population of the USA is 3.3\times10^{8} , rounded to two significant figures.

Write this distance as an ordinary number.

(b) The population of Washington DC is 690000 rounded to two significant figures.

Write this number in standard form.

(a)  3 30000000 km

(b)  6.9\times10^{5}km

3.   Put the below numbers in order. Start with the smallest number.

0.092 \quad \quad 2.9\times10^{-3} \quad \quad 0.00029 \quad \quad 209\times10^{-4}

Converting all of the numbers to the same form or standard notation for comparison or 3 of the four numbers ordered correctly.

0.00029, \quad \quad 2.9\times10^{-3} , \quad \quad 209\times10^{-4}, \quad \quad 0.092

Learning checklist

You have now learned how to:

• Convert ordinary numbers standard form
• Convert standard form to ordinary numbers
• Adjust numbers to standard form notation

The next lessons are

• Linear equations
• Quadratic equations

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Standard Form

What is "Standard Form"?

that depends on what you are dealing with!

I have gathered some common "Standard Form"s here for you..

Note: Standard Form is not the "correct form", just a handy agreed-upon style. You may find some other form to be more useful.

Standard Form of a Decimal Number

In Britain this is another name for Scientific Notation , where you write down a number this way:

In other countries it means "not in expanded form" (see Composing and Decomposing Numbers ):

Standard Form of an Equation

The "Standard Form" of an equation is:

(some expression) = 0

In other words, "= 0" is on the right, and everything else is on the left.

Example: Put x 2 = 7 into Standard Form

x 2 − 7 = 0

Standard Form of a Polynomial

The "Standard Form" for writing down a polynomial is to put the terms with the highest degree first (like the "2" in x 2 if there is one variable).

Example: Put this in Standard Form:

3 x 2 − 7 + 4 x 3 + x 6.

The highest degree is 6, so that goes first, then 3, 2 and then the constant last:

x 6 + 4 x 3 + 3 x 2 − 7

Also, within each term, it is nice to have the variables in alphabetical order (if it does not make things more confusing):

yzx 2 + 4 yx 3

The highest degree is 3, so that goes first, also put the variables in alphabetical order

4 x 3 y + 3 x 2 yz

Standard Form of a Linear Equation

The "Standard Form" for writing down a Linear Equation is

Ax + By = C

A shouldn't be negative, A and B shouldn't both be zero, and A , B and C should be integers.

Bring 3x to the left:

−3x + y = 2

Multiply all by −1:

3x − y = −2

Note: A = 3, B = −1, C = −2

Ax + By + C = 0

is sometimes called "Standard Form", but is more properly called the "General Form".

Standard Form of a Quadratic Equation

The "Standard Form" for writing down a Quadratic Equation is

( a not equal to zero)

Expand "x(x−1)":

x 2 − x = 3

Bring 3 to left:

x 2 − x − 3 = 0

Note: a = 1, b = −1, c = −3

Standard Form of a Circle Equation

With a circle like this:

The Standard Form is this:

(x−a) 2 + (y−b) 2 = r 2

See Circle Equations for more details.

Problem solving with standard form

I can use my knowledge of standard form to solve problems.

Lesson details

Key learning points.

• Standard form is used frequently in Science.
• The exponent can tell you how large the number will be.
• The exponent can tell you how small the number will be.

Common misconception

Pupils can incorrectly write a number in standard form or use a number in incorrect standard form whereby the number A does not satisfy 1 ≤ A < 10 or pupils use division of positive powers of 10.

Standard form represents a multiplicative relationship, so there should always be a multiplication. Embedding the understanding that negative exponents refer to 1/10^n is important. Using the place value chart with fractional and exponent form helps.

Standard form - Standard form is when a number is written in the form A × 10^n, (where 1 ≤ A < 10 and n is an integer).

Associative law - The associative law states that a repeated application of the operation produces the same result regardless of how pairs of values are grouped. We can group using brackets.

This content is © Oak National Academy Limited ( 2024 ), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Starter quiz

6 questions.

Standard Form of a Linear Equation: Explanation, Review, and Examples

• The Albert Team
• Last Updated On: March 1, 2022

Keep reading to review the standard form of a linear equation. We’ll learn why we use the standard form of linear equation as well as how to write equations and graph with the standard form. Lastly, we will review other forms of linear equations.

Using our hands, we can change a piece of clay into a work of art. Likewise, using our mathematical tools, we can change an equation into a different form. The different forms provide us with useful information.

Now, let’s dive into standard form!

What We Review

What is the standard form of a linear equation?

The standard form of a linear equation, also known as the “ general form “, is:

The letters a , b , and c are all coefficients. When using standard form, a , b , and c are all replaced with real numbers. The letter x represents the independent variable and the letter y represents the dependent variable.

A few notes on Standard Form:

• The a term must be a positive integer
• a, b, and c cannot be decimals or fractions

Return to the Table of Contents

Why use standard form?

The standard form of a line can be particularly helpful when solving a system of equations. For instance, when using the elimination method to solve a system of equations, we can easily align the variables using standard form. 

System of equations with standard form

Let’s see a quick example. If we were given the system of equations:

…we can rewrite the equations in standard form.

Then, we can solve using the elimination method by multiplying the second equation by 4 .

By adding these equations together we obtain: 9y=49 . When we solve that, we know y=\frac{49}{9} .

Then, we can substitute the value of y into one of the original equations to determine the value of x .

Now we have solved the system of equations. The solution is (\frac{8}{9},\frac{49}{9}) . Using standard form allowed us to use the elimination method to solve the system.

As we’ll see below , standard form is also useful for easily determining the intercepts of a linear function.

How to write a linear equation in standard form (example)

Let’s write an equation of the line with a slope of 4 and a y -intercept of 7 in standard form.

To begin, we will first write the equation in slope-intercept form. This is the easiest form to write when given the slope and the y -intercept.

We know the slope, m , is 4 and the y -intercept, b , is 7 .  

To change this into standard form, all we need to do is subtract the x term from both sides, in this case 4x .

We have now written the standard from of a linear equation. The linear equation with a slope of 4 and a y -intercept of 7 is y-4x=7 .

Are you more of a visual learner? Checkout the video below with another example of writing linear equations in standard form:

How to graph a standard form linear equation (example)

We also need to know how to graph a standard form equation. In standard form, we can easily determine the x and y -intercepts.

Let’s graph the equation 3y-5x=30 .

Find x-intercept

First, we can determine the x -intercept. Remember, this is where the line crosses the x -axis and where y=0 . To do so, we will substitute 0 for y .

Therefore, the x -intercept is at -6 . This means the point (-6,0) is on the graph.

Find y-intercept

Let us now determine the y -intercept. Remember, this is where the line crosses the y -axis and where x=0 . To do so, we will substitute 0 for x .

Therefore, the y -intercept is at 10 . This means the point (0,10) is on the graph.

Draw the graph

We now plot the x and y -intercepts. We are plotting the points (-6,0) and (0,10) .

The very last step is simply to connect the points on the graph. This creates the graph of the standard form equation 3y-5x=30 .

Now you know how to graph a standard form equation!

To view another example of graphing from standard form, check out the video below:

Other forms of linear equations

Slope-intercept form.

Linear equations can be written in slope-intercept form, determined by the slope and the y -intercept of a line. For more info, visit our review guide on slope-intercept form.

Point-Slope Form

A linear equation can also be written in point-slope form . This form is determined by one point and the slope of the line. For more details, read our point-slope review guide .

Summary: Standard Form

In this review post, we’ve learned:

• Standard form of an equation is: ax+by=c
• Standard form is useful for solving systems of equations and for determining intercepts
• How to write a linear equation in standard form
• How to graph an equation in standard form

Click here to explore more helpful Albert Algebra 1 review guides .

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Standard Form: Lessons

If you want a complete lesson, a Tarsia jigsaw, or a fun and engaging lesson activity, then you have come to the right place!

Complete Lesson PowerPoints keyboard_arrow_up Back to Top

Complete lessons from some of my favourite authors. There are few better places to your own planning process.

Tarsia Jigsaws and Card Sorts keyboard_arrow_up Back to Top

I have been a Tarsia fan for many years. Visit the my tarsia page for lots of different ideas for using this wonderful free resource, and to get hold of the software yourself.

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A selection of some of my favourite, free maths activities to use in lessons. I have tried out each and every one with my students.

Standard Form

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Standard Form Equation of a line

Examples of equation and Graphs

Overview of different forms of a line's equation

There are many different ways that you can express the equation of a line . There is the slope intercept form , point slope form and also this page's topic. Each one expresses the equation of a line, and each one has its own pros and cons. For instance, point slope form makes it easy to find the line's equation when you only know the slope and a single point on the line. Standard form also has some distinct uses, but more on that later.

Definition of Standard form Equation

$ \text{Formula } : \\ Ax + By = C \\ A \ne 0 \\ B \ne 0 $

General Formula for x and y-intercepts

For the equation of a line in the standard form, $$ Ax + By = C $$ where $$ A \ne 0 $$ and $$ B \ne 0$$ , you can use the formulas below to find the x and y-intercepts.

$ \text{X Intercept: } \\ \frac C A = \frac 6 3 = 2 $

$ \text{Y Intercept: } \\ \frac C B = \frac 6 2 = 3 $

Example and Non Example Equations

When is standard form useful?

• When you want to graph the line.
• When you know the y-intercept of the line .
• Contrast this with slope intercept form -- in this case, you have to do more work to find the x-intercept .

When are other forms more useful?

• Slope intercept form and point slope make it easier to find the slope of your line . In standard form, you must do some work to get the slope .
• Point slope makes it easy to graph the line when you only know the line's slope and a single point or when you know 2 points on the line.

Video Tutorial on Standard Form Equation of a Line

Sample Practice Problems

Find the intercepts and graph the following equation: 3x + 2y = 6

How to find the x-intercept :

How to find the y-intercept :

How to Graph from Standard Form

Plot the x and y-intercepts and draw the line on the graph paper!

Practice Problems

Identify which equations below on the right are in standard form.

• Equation 1: 2x + 5 = 2y
• Equation 2 : 2x + 3y = 4
• Equation 3: y = 2x + 3
• Equation 4 : 4x -$$ \frac 1 2 $$ y = 11

Equation 2 and equation 4 are the only ones in standard form.

Equation 3 is in Slope intercept form .

• Equation 1 : 11 = ¼x + ½y
• Equation 2: 2x + 5 + 2y = 3
• Equation 3: y - 2 = 3(x − 4)
• Equation 4 : $$ \frac 1 2 $$ y − 4x = 0

Equation 1 and equation 4 are the only ones in standard form.

Equation 3 is in point slope form .

Find the intercepts and then graph the following equation 2x + 3y = 18 .

First, find the intercepts by setting y and then x equal to zero. This is pretty straightforward since the line is already in standard form.

Set y = 0 :

Solve for x:

Solve for y:

Plot the x and y-intercepts , which in this case is (9, 0) and (0, 6) and draw the line on the graph paper!

Find the intercepts and then graph the following equation 3x + 5y = 15 .

Plot the x and y-intercepts , which in this case is (5, 0) and (0, 3) and draw the line on the graph paper!

Find the intercepts and then graph the following equation 3y − 2x = -12 .

Plot the x and y-intercepts , which in this case is (6, 0) and (0,-4) and then graph the equation!

• Linear Equations
• Equation of Line Formula
• Slope Intercept Form
• Slope Intercept to Standard Form
• Point Slope to Standard Form
• Worksheet on standard form equation (pdf with answer key on this page's topic)

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GCSE 9-1 Exam Question Practice (Standard Form)

Subject: Mathematics

Age range: 14-16

Resource type: Lesson (complete)

Last updated

17 January 2019

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Standard Form – Definition with Examples

What is standard form, examples of standard form of numbers, real-life examples of standard form, solved examples of standard form, practice problems of standard form, frequently asked questions of standard form.

Did you know that Earth is 4,543,000,000 years old? We know you skipped past all those zeroes, so we will give you a simpler way of saying it: Earth is 4.543 billion years old. Do you see how reading numbers a certain way makes them easier to understand? That’s why all the mathematicians in the world decided to agree on some rules on writing mathematical concepts so that it is convenient for everyone to read, write, and work with. This particular way is called the standard form.

All of the things you see in math, like numbers or fractions or equations or expressions , have a standard form defined for them. We can think of the standard form as the most common way of representing a mathematical element . 

Let’s look at the standard form of some common mathematical elements:

Whole Numbers

You can define the standard form of a whole number as follows.

Any number that we can write as a decimal number , between 1.0 and 10.0, multiplied by a power of 10, is said to be in standard form.

For example, take the number 123,000,000; an easier way of writing this number is ​​

1.23 × 10 8 ;

If you observe carefully, 1.23 is a decimal number between 1.0 and 10.0 and so we have the standard form of 123,000,000 as 1.23 × 10 8 .

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• 14,300,000 in the standard form is 1.43 × 10 7 .
• 3000 in the standard form is 3 × 10³.
• Some other examples are 1.98 ✕ 10¹³ and  0.76 ✕ 10¹³.

Factoid: “Standard Form” is also called “Scientific Notation” depending on the mathematical lingo of the country you are in. In the United Kingdom and the countries that follow the same conventions as the United Kingdom, the term “Scientific Notation” is most commonly used whereas in countries that follow the US conventions majorly refer to this form of writing as the “Standard Form”.

Example 1: Consider the number 81,900,000,000,000.

Step 1: Write the first number: 8 .

Step 2: Add a decimal point after this and write the remaining non-zero numbers: 8.19

Step 3: Now count the number of digits after 8. There are 13 digits. This 13 will be the power of 10 while writing the given number in standard form.

Step 4: So, in standard form: 81,900,000,000,000 is 8.19 × 10¹³.

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More Worksheets

• The distance between the Sun and Mars is 141,700,000 miles or 228,000,000 km.

         This distance can easily be written in standard form as: 1.417 × 108 miles or 2.28 × 108 km.

• Atoms are tiny units of matter composed of three fundamental particles—proton, neutron, and electron. A proton and a neutron weigh equally, 1.67 × 10–27 kg. The weight of an electron is 9.11 × 10–31 kg.

Many other quantities, such as the size of planets, the size of microorganisms, the size of microchips, and the country’s population are all expressed in standard form.

A fraction is said to be in standard form when both the numerator and denominator are co-prime numbers.

Factoid: Two numbers are said to be co-prime when their only common divisor(or factor ) is 1. For example, 2 and 3, 4 and 9, 6 and 13. By virtue, two prime numbers are always co-prime.

For example, the fraction  ⅚  is a standard fraction because the numerator 5 and denominator 6 do not have only 1 as their common divisor , whereas the fraction 4/8 is not a standard fraction because the numerator and denominator have common divisors other than 1, i.e., 2 and 4. But no worries, do you want to turn a fraction into its standard form? We got your back. Let’s look at an example.

Consider the fraction 12/20. Step 1: Find all the common divisors of the numerator and the denominator.             In this case, 12 and 20 have common divisors: 1,2,4.

Step 2: Divide the numerator and denominator by their greatest common divisor. Like this:

No Step 3 is required! As you can see ⅗ is a standard fraction and is the standard form of 12/20.

Examples of Standard Form of Fractions

• The most famous standard fraction is 22/7, also called “pi”.
• All unit fractions , i.e., fractions with numerator 1 are standard.

Decimal Numbers

The definition for the standard form of decimal numbers is the same as that of whole numbers,

Any number that we can write as a decimal number, between 1.0 and 10.0, multiplied by a power of 10, is said to be in standard form.

The scoop about the standard form of decimal numbers lies in their steps. This is a bit different from whole numbers, where you may have noticed, the power of 10 was a positive number like in 1.23 × 10 8 . Let’s see how we can convert a scary-looking decimal number into a pretty-looking decimal number.

Consider the decimal number 0.0004789.

Step 1: Make your decimal number jump from its original position and place it after the first non-zero digit: 4.789 .

Step 2: Count the number of digits you jumped. In this case, we jumped 4 digits. This number will be the power of 10 while writing our decimal number in standard form.

Step 3: If you have made jumps in the direction to the right, the power of 10 will be negative. If you made the jumps in the direction to the left, the power of 10 will be positive.

Step 4: So, in standard form: 0.0004789 is 4.789 × 10 -4 .

Using the same method, the standard form of the decimal number 981.23 will be 

9.8123 × 10 2 .

The standard form of mathematical elements allows us to work in a convenient and efficient manner by making them easily readable for everyone. There is no shortcut to finding the standard form of anything, but here are a few tips and tricks that we have summarized for you that you must keep in mind while writing something in standard form.

Tips to Master Standard Form

• Thoroughly understand the two parts of the standard form: digits and the power of 10.
• Remember that the power is negative if you have made jumps to the right and the power is positive when you have made jumps to the left while putting a decimal number in its standard form.
• Count the places you have moved the decimal point twice before writing the final answer.
• Don’t forget about the second cousin of prime numbers: the co-primes. They are the basics of putting a fraction in its standard form.

Express 321,000,000 in scientific notation.

The scientific form of the number is 3.21 x 10 8 .

Express 0.00005432 in the standard form.

The scientific form of the number is 5.432 x 10 -5 .

Write 25/40 in the standard form.

The scientific form of the given fraction is ⅝.

Standard Form

Attend this Quiz & Test your knowledge.

What is the standard form of the number 78,980,000?

What is the standard form of the number 32145.222, what is the standard form of the fraction 12/18.

Is the standard form the same as a decimal form?

Yes. The standard form, decimal form, and scientific notation of a number are the same.

How do you convert a fraction into its standard form?

A fraction can be converted into standard form by dividing the numerator and denominator by their greatest common divisor.

What is the basic rule of writing standard form?

While writing decimals in scientific notation, standard form, or decimal form, move the decimal place to the left or right until you reach a number from 1-10.

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Problem Solving, Using and Applying and Functional Mathematics

Problem solving.

The problem-solving process can be described as a journey from meeting a problem for the first time to finding a solution, communicating it and evaluating the route. There are many models of the problem-solving process but they all have a similar structure. One model is given below. Although implying a linear process from comprehension through to evaluation, the model is more of a flow backward and forward, revisiting and revising on the problem-solving journey.

Comprehension

Representation.

• Can they represent the situation mathematically?
• What is it that they are trying to find?
• What do they think the answer might be (conjecturing and hypothesising)?
• What might they need to find out before they can get started?

Planning, analysis and synthesis

Having understood what the problem is about and established what needs finding, this stage is about planning a pathway to the solution. It is within this process that you might encourage pupils to think about whether they have seen something similar before and what strategies they adopted then. This will help them to identify appropriate methods and tools. Particular knowledge and skills gaps that need addressing may become evident at this stage.

Execution and communication

During the execution phase, pupils might identify further related problems they wish to investigate. They will need to consider how they will keep track of what they have done and how they will communicate their findings. This will lead on to interpreting results and drawing conclusions.

Pupils can learn as much from reflecting on and evaluating what they have done as they can from the process of solving the problem itself. During this phase pupils should be expected to reflect on the effectiveness of their approach as well as other people's approaches, justify their conclusions and assess their own learning. Evaluation may also lead to thinking about other questions that could now be investigated.

Using and Applying Mathematics

Aspects of using and applying reflect skills that can be developed through problem solving. For example:

In planning and executing a problem, problem solvers may need to:

• select and use appropriate and efficient techniques and strategies to solve problems
• identify what further information may be required in order to pursue a particular line of enquiry and give reasons for following or rejecting particular approaches
• break down a complex calculation problem into simpler steps before attempting a solution and justify their choice of methods
• make mental estimates of the answers to calculations
• present answers to sensible levels of accuracy; understand how errors are compounded in certain calculations.

During problem solving, solvers need to communicate their mathematics for example by:

• discussing their work and explaining their reasoning using a range of mathematical language and notation
• using a variety of strategies and diagrams for establishing algebraic or graphical representations of a problem and its solution
• moving from one form of representation to another to get different perspectives on the problem
• presenting and interpreting solutions in the context of the original problem
• using notation and symbols correctly and consistently within a given problem
• examining critically, improve, then justifying their choice of mathematical presentation
• presenting a concise, reasoned argument.

Problem solvers need to reason mathematically including through:

• exploring, identifying, and using pattern and symmetry in algebraic contexts, investigating whether a particular case may be generalised further and understanding the importance of a counter-example; identifying exceptional cases
• understanding the difference between a practical demonstration and a proof
• showing step-by-step deduction in solving a problem; deriving proofs using short chains of deductive reasoning
• recognising the significance of stating constraints and assumptions when deducing results
• recognising the limitations of any assumptions that are made and the effect that varying the assumptions may have on the solution to a problem.

Functional Mathematics

Functional maths requires learners to be able to use mathematics in ways that make them effective and involved as citizens, able to operate confidently in life and to work in a wide range of contexts. The key processes of Functional Skills reflect closely the problem solving model but within three phases:

• Making sense of situations and representing them
• Processing and using the mathematics
• Interpreting and communicating the results of the analysis