representation of number meaning

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Pythagoreanism

• Nature’s numbers
• Number’s nature

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The earliest known systematic cult based on the rule of numbers was that of the Pythagoreans . Pythagoras was a Greek who thrived in the 6th century bce . Little is known of his life, and in fact he may be a composite figure to whom the discoveries of many different people have been attributed by his followers. It is not even known whether the Pythagorean theorem in geometry was actually discovered by him.

The Pythagoreans invested specific numbers with mystical properties. The number 1 symbolized unity and the origin of all things, since all other numbers can be created from 1 by adding enough copies of it. For example, 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1. The number 2 was symbolic of the female principle, 3 of the male; they come together in 2 + 3 = 5 as marriage. All even numbers were female, all odd numbers male. The number 4 represented justice . The most perfect number was 10 , because 10 = 1 + 2 + 3 + 4. This number symbolized unity arising from multiplicity. Moreover, it was related to space. A single point corresponds to 1, a line to 2 (because a line has two extremities), a triangle to 3, and space to 4. Thus 10 also symbolized all possible spaces.

The Pythagoreans recognized the existence of nine heavenly bodies: Sun, Moon, Mercury, Venus, Earth, Mars, Jupiter, Saturn, and the so-called Central Fire. So important was the number 10 in their view of cosmology that they believed there was a tenth body, Counter-Earth, perpetually hidden from us by the Sun.

Some Pythagorean speculations were mathematical. They represented numbers by arrangements of dots. The square numbers (1, 4, 9, 16,…) were arranged in squares, and the triangular numbers (1, 3, 6, 10,…) were arranged in triangles ( see figure ). This terminology remains in use to the present day.

The Pythagoreans were especially fascinated by the presence of numbers in the natural world. Perhaps their most spectacular discovery was that musical harmony is related to simple whole-number ratios. A string (such as that on a violin) produces a note with a particular pitch ; a string one-half as long produces an extremely harmonious note to the first, now called the octave. A string two-thirds as long produces the next most harmonious note, now called the fifth. And one three-fourths as long produces the fourth, also very harmonious. The Pythagoreans discovered these facts empirically by experimenting with strings of different lengths. Today these harmonies are traced to the physics of vibrating strings, which move in patterns of waves. The number of waves that can fit into a given length of string is a whole number, and these whole numbers determine the simple numerical ratios. When the numbers do not form a simple ratio, the corresponding notes interfere with each other and form discordant “beats” that are unpleasant to the ear. The full story is more complex, involving what the brain becomes accustomed to, but there is a definite rationale behind the Pythagorean discovery. This later led the German astronomer Johannes Kepler to the concept of the “music of the spheres,” a kind of heavenly harmony in which the planets effectively produced tunes as they moved across the heavens. Some of Kepler’s theories about the planets, such as the elliptical shape of their orbits, became solid science—but not this one. Nonetheless, it was influential in establishing the view that there is some kind of order in the cosmos, an idea that culminated in Isaac Newton ’s law of gravity .

Cultural associations of some numbers

The enormous range of symbolic roles that numbers have played in various cultures , religions, and other systems of human thought can be gauged from a brief sample.

Not surprisingly, the number 1 is generally treated as a symbol of unity. Therefore, in monotheistic religions, it often symbolizes God or the universe. The Pythagoreans did not consider 1 to be a number at all because number means plurality and 1 is singular. However, they considered it to be the source of all numbers because adding many 1s together can create any other (positive whole) number. In their system, where odd numbers were male and even numbers female, the number 1 was neither; instead, it changed each to the other. If 1 is added to an even number, it becomes odd; similarly, if 1 is added to an odd number, it becomes even.

The number 2 symbolizes many of the basic dualities: me/you, male/female, yes/no, alive/dead, left/right, yin/yang , and so on. Dualities are common in human approaches to the world, probably because of our preference for two-valued logic—yet another duality, true/false. Although 2 was female to the Pythagoreans, other numerological schemes viewed it as male. In Agrippa von Nettesheim ’s De occulta philosophia (1533; “On the Philosophy of the Occult”), 2 is the symbol for man, sex, and evil. One reason that some have associated 2 with evil is that the biblical book of Genesis does not use the formula “and it was good” when referring to the second day of Creation.

Some religions are dualistic, with two gods in place of the one God of monotheism. Examples include Zoroastrianism , where Ahura Mazdā (the god of light and goodness) battles with Ahriman (the god of darkness and evil). The number 2 is often associated with negatives, as in the words duplicity and two-faced . Northwest Coast Indian s required the parents of twins to observe various taboos because they believed that supernatural powers would bring the wishes of twins to fruition.

The number 3 is a very mystical and spiritual number featured in many folktales (three wishes, three guesses, three little pigs, three bears, three billy goats gruff). In ancient Babylon the three primary gods were Anu , Bel ( Baal ), and Ea , representing Heaven, Earth, and the Abyss. Similarly, there were three aspects to the Egyptian sun god: Khepri (rising), Re (midday), and Atum (setting). In Christianity there is the Trinity of God the Father, God the Son, and God the Holy Spirit . Plato saw 3 as being symbolic of the triangle , the simplest spatial shape, and considered the world to have been built from triangles. In German folklore a paper triangle with a cross in each corner and a prayer in the middle was thought to act as protection against gout, as well as protecting a cradle from witches. Three black animals were often sacrificed when attempting to conjure up demons. On the other hand, a three-coloured cat was a protective spirit. In William Shakespeare ’s Macbeth (1606–07) there are three witches, and their spell begins, “Thrice the brindled cat hath mewed,” reflecting such superstitions. Also, 3 is the dimension of the smallest magic square in which every row, column, and diagonal sums to 15.

The number of order in the universe is 4—the four elements of earth, air, fire, and water; the four seasons; the four points of the compass; the four phases of the Moon (new, half-moon waxing, full, half-moon waning). The Four Noble Truths epitomize Buddhism. To the Pythagoreans 4 was the source of the tetractys 1 + 2 + 3 + 4 = 10, the most perfect number . In medieval times there were thought to be four humour s (phlegm, blood, choler, and black bile—hence the adjectives phlegmatic , sanguine , choleric , and melancholic ), and the body was bled at various places to bring these humours into balance.

The number 4 is central in the world view of the Sioux , with four groups of gods (superior, ally, subordinate, and spirit), four types of animal (creeping, flying, four-legged, and two-legged), and four ages of humans (infant, child, mature, and elderly). Their medicine men instructed them to carry out all activities in groups of four.

Because 4 is generally a practical, material number, few superstitions are associated with it. An exception is in China, where 4 is unlucky because she (“four”) and shi (“death”) sound similar. In the biblical Revelation to John the Four Horsemen of the Apocalypse wreak destruction upon humanity.

The sum of the first even and odd numbers (2 + 3) is 5. (To the Pythagoreans 1 was not a number and was not odd.) It therefore symbolizes human life and—in the Platonic and Pythagorean traditions—marriage, as the sum of the female 2 and the male 3. The Pythagoreans discovered the five regular solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron; now known as the Platonic solid s). Early Pythagoreanism acknowledged only four of these, so the discovery of the fifth (the dodecahedron, with 12 pentagonal faces) was something of an embarrassment. Perhaps for this reason 5 was often considered exotic and rebellious.

The number 5 was associated with the Babylonian goddess Ishtar and her Roman parallel, Venus , and the symbol for both was the five-pointed star, or pentagram . In England a knot tied in the form of the pentagram is called a lover’s knot because of this association with the goddess of love. In Manichaeism 5 has a central position: the first man had five sons; there are five elements of light (ether, wind, water, light, and fire) and a further five of darkness. The body has five parts; there are five virtues and five vices.

The number 5 was also important to the Maya , who placed a fifth point at the centre of the four points of the compass. The five fingers of the human hand lent a certain mystery to 5, as did the five extremities of the body (two arms, two legs, head). A human placed in a circle with outspread arms and legs approximates the five points of a pentagon, and if each point is joined to its second nearest neighbour, a pentagram results. This geometric figure is central to occultism, and it plays a prominent role in summoning spells whereby it is supposed to trap a demon, or devil, who can then be compelled to do the sorcerer’s bidding. The belief that 5 was sacred led to an extra element, augmenting the traditional four that made a human being . This fifth essence, or quintessence , is the origin of the word quintessential .

In Islam 5 is a sacred number. Foremost are the five Pillars of Islam : declaration of faith ( shahādah ), prayer ( ṣalāt ), fasting during Ramadan , giving alms ( zakāt ), and making the pilgrimage to Mecca (the hajj ). Prayers are said five times every day. There are five categories of Islamic law and five law-giving prophets ( Noah , Abraham , Moses , Jesus , and Muhammad ).

By a wonderful conjunction of mathematical coincidences, 6 is both the sum (1 + 2 + 3) and the product (1 × 2 × 3) of the first three numbers. It is therefore considered “perfect.” In mathematics, a perfect number is one that equals the sum of its divisors (excluding itself), and 6 is the first perfect number in this sense because its divisors are 1, 2, and 3. The next perfect number is 28. No odd perfect numbers are known, but it has not been proved that none exists. The perfection of 6 shows up in the six days of Creation in Genesis , with God resting on the seventh day. The structure of the Creation parallels the sum 1 + 2 + 3: on day 1 light is created; on days 2 and 3 heaven and earth appear; finally, on days 4, 5, and 6 all living creatures are created.

Numerals – Definition, Numeral Systems, Facts,  Examples, FAQs

What are numerals, numbers vs. numerals, what are different numeral systems, solved examples on numerals, practice problems on numerals, frequently asked questions about numerals.

Numerals are symbols or signs (or collection of symbols) used to represent numbers corresponding to numerical quantities, values, or measurements.  

Numerals often come together with a set of rules for representing larger numbers. They provide a concise and standardized way to express numerical quantities. 

Numerals help us communicate numeric values or numerical quantities effectively. In math, a number conveys a quantity. You can say that a number offers an answer to the question “how many things are there?.” A numeral helps us to write it down.

Example 1: If you have 3 apples, you can use different numerals to represent this quantity.

• Number: 3 
• Number name: Three 
• Roman numeral: III
• Symbol/Sign (using dots):  ●●●

Example 2: Take a look at the image given below. What do you see?

This image shows two cups, two soccer balls, and two ice cream cones. Here, “two” indicates a quantity. We can represent this quantity using different numerals, such as 

• Word form: Two
• Roman numerals: II 
• Using dots: ●●

So, each of these is a numeral representing the quantity “two.”  

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Numerals: Definition

A numeral is a symbol or a sign that stands for a number. A numeral can be defined as a representation of a number, which can be in the form of a figure, symbol, name, or a combination of figures and symbols.

Example : Seven, VII, ७, etc. are all symbols or signs we use to indicate the number “7.” 

Each one of them is a numeral.  

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The terms “numbers” and “numerals” can be confusing. The concept of a number is distinct from its written representation, which is the numeral. Let’s understand the difference.

A numeral system is a way to represent numbers. Numeral systems, also known as number systems, indeed consist of symbols and rules for representing numbers and quantities. They play a crucial role in mathematics and are used to express numerical values in a standardized and systematic manner.

There are many ways to represent a number with the help of symbols in math. In other words, we can represent a number using more than one numeral.

Note that many numeral systems have evolved across various regions and cultures, each serving a distinct purpose. Let’s discuss a few.

• Arabic numerals or Indo-Arabic numerals (Hindu-Arabic numerals):

Arabic numerals, also known as Indo-Arabic or Hindu-Arabic numerals, are the ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) we commonly use in our number system. The decimal number system is the extension of the Indo-Arabic numeral system.

• Roman numerals:  

Roman numerals are a numeral system originating from ancient Rome, characterized by the use of letters from the Latin alphabet to represent numbers. Roman numerals are still used today in specific contexts, such as numbering book chapters or indicating the date on clock faces.

I (1), II (2), V (5), X (10), L (50), C (100), D (500), M (1000), etc.

• Binary numerals:  

Binary numerals, often referred to as base-2 numerals, are a numeral system that uses only two digits: 0 and 1. It is widely used in computer systems.

• Octal numerals: 

Octal numerals, a base-8 numeral system, use eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. 

• Chinese numerals:

Chinese numerals, also known as Han numerals, are used in the Chinese writing system to represent numbers. 

$\bigcirc$ (0), 一 (1), 二 (2), 三 (3), 四 (4), 五 (5), 六 (6), 七 (7), 八 (8), 九 (9), 十 (10)

• Number names or Word form : 

• Devanagari numerals: 

Devanagari numerals are the numeric symbols used in the Devanagari script, primarily in the Indian subcontinent. They are the scripts used for languages like Sanskrit, Hindi, Marathi, and Nepali.

• Tally marks : 

Tally marks are a simple system of counting and recording numbers using vertical lines or strokes, in sets of 5. Each group of five tally marks is typically represented by four vertical lines, with the fifth line being drawn diagonally across the previous four. 

What Is the Decimal Numeral System?

The decimal numeral system is the most commonly used number system. This system is also known as the base-10 positional numeral system since it used 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to form all the numbers. It is derived from the Indo-Arabic numeral system.

Decimal numerals (often called simply as decimals refers to the notation of a number. In this system, the integer portion and the decimal portion of a numeral are separated using a decimal point. 

• What Are Place Values?

Place value refers to the value of a digit in a number by the virtue of the position in the number. The place values of digits in the decimal system are represented using the powers of 10. In decimal numbers, the whole number part and the fractional part is separated by the decimal point.

In the international place value chart, there are three periods for the nine places from right to left. Each period has three place values.

Facts about Numerals

• Numerals are symbols used to represent numbers. Different numeral systems are used worldwide, reflecting cultural and historical diversity.
• The Roman numeral system indeed did not have a symbol for zero.
• The ancient Egyptians made significant contributions to the development of numerals and mathematical systems.
• The earliest numerals used by humans were simple scratches on a rock or stone, notches on a stick, etc.

In this article, we learned about numerals. We explored a few commonly known numeral systems as well. To deepen our understanding, let’s dive into practical examples and MCQs for enhanced comprehension.

Example 1: What is the word form for 42?   

Solution : 

The word form for the number 42 is “Forty Two.” 

Example 2: How many candies are there in the bowl? Represent the numerical quantity using different numerals.    

There are 9 candies in the bowl.

Example 3: What is the time shown in the given analog clock?

The hour hand points exactly to V, which is the roman numeral for 5. 

The minute hand points exactly to XII, which is the roman numeral for 12.

So, the clock reads 5 o’clock.   

Numerals - Definition, Numeral Systems, Facts,  Examples, FAQs

Attend this quiz & Test your knowledge.

What is the roman numeral for 4?

Which of the following is not a numeral for 1, in the decimal numeral system, the base is ____., identify the numeral of 7..

What are digits in math?

A digit is a single numerical symbol used to represent numbers. A number is a mathematical concept or idea used to represent a quantity.

What is the difference between a numeral and a digit?

A numeral is a symbol or group of symbols used to represent a number, either using digits, words, other signs and symbols, etc.

What is the difference between digits, numbers, and numerals?

Number is expressed with digits, while a numeral is how we describe a number.

What is the place value of a digit?

The place value of a digit indicates its position in a number. For example, in 36, the place value of 3 is 30, while that of 6 is 6.

Are all numeral systems based on place values?

No, all numeral systems are not based on place values. For instance, the Roman numeral system is not based on place values.

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