Number

Pronunciation: /ˈnʌm bər/ ?

A number is a representation of a quantity.[1] A quantity answers the questions, "How many?" and "How much?"

3 apples
Figure 1: 3 apples
In figure 1 there are three apples. We can use different characters to represent the quantity:
3
three
3.0
6/2
4-1
square root of 9
11 in base 2
Table 1: Representations of the quantity 3
How many ways can you think of to represent 3?

Related Words

Numeric
Numerical
Numerically
Quantity

Digits and Place Value

The digits of the number 374 is 3 in the hundred's place, 7 in the ten's place and 4 in the ones place.
Figure 2: Digits and place value

Each digit of a number has a place value. The position of the digit in the number tells the place value. For the decimal numbers, the place values are multiple of 10. Figure 2 shows a number without a decimal point. This means that the digit on the extreme right of the number has a place value of 1. The total value of the digit is 4 · 1 = 4. The digit following to the left is '7'. The place value of this digit is 1 · 10 = 10. The total value of the digit is 10 · 7 = 70. The digit to the left has a place value of 10 · 10 = 100. The total value of the digit is 3 · 100 = 300. The value of the number in figure 2 is 300 + 70 + 4.

The digits of the number 3.682, 6 is in the tenth's place, 8 is in the hundredth's place, and 2 is in the thousandth's place
Figure 3: Digits and place value to the right of the decimal point.

In figure 3, number 3.682 has a decimal point. The digit to the left of the decimal has a place value of 1. The digit to the right of the decimal has a place value of 1/10. Each digit further to the right of the decimal point has a place value of 1/10 of the previous digit. So the 6 has a total value of 6 · 1/10 = 6/10. The following digits have values of 8 · 1/100 = 8/100 and 2 · 1/1000. The value of the integral number is 3 + 6/10 + 8/100 + 2/1000.

Parts of a Number

Parts of a decimal number: sign, whole part, decimal, part less than one
Figure 4: Parts of a number.
A number can consist of digits, a decimal point, a positive or negative sign (+ or -), and a comma separator.

Other Representations of Quantity

Seven hash marks. Each has mark is a vertical line segment.
Figure 5: Seven hash marks
Quantities have been represented in many ways. One way that was in early use was hash marks. Each line counts as one. In the example in figure 5, there are seven lines. So the quantity represented is seven.

Notched bone from Turkey
Figure 6: A notched bone, which may have been used as a musical instrument or a counting device, was found in an Early Bronze Age (2900 B.C.) grave at Kenan Tepe, Turkey.
Photographer: Bradley J. Parker
Sticks and bones have been found at archaeological sites with notches cut in them to make hash marks. Hash marks have several problems. Zero can not be represented by hash marks. Also, very large numbers are awkward to represent with has marks. Only natural numbers can be represented by hash marks.

Brass counters and Counting boards were used to calculate numbers before calculators were invented. Figure 7 shows brass counters used at Jamestown.

Brass counter markers excavated from Jamestown
Figure 7: Brass counters excavated from Jamestown.

Numbers have evolved as the best way to represent quantities. They can represent positive numbers, zero, and negative numbers. They can represent very large and very small numbers. They can also represent non-whole quantities such as 1.5. The only thing numbers can not represent exactly is irrational quantities. The digits in irrational values go on forever without repeating, so an irrational value can not be represented with a finite number of digits. Other representations such as π and 2 are used to represent irrational numbers exactly.

Types of Numbers

Venn diagram of number types.
: Complex number
: Real number
Double struck I: Imaginary number
: Rational number
: Integer
: Natural number (counting numbers)
Figure 8: Number Types

Mathematicians have divided numbers into groups according to their properties. Figure 8 is a Venn diagram of number types. The group of complex numbers contains all other groups of numbers. All numbers are complex numbers. Table 2 gives the properties of the types of numbers.

Types of Numbers
TypeSymbolDescriptionExamples
Complex Number A complex number is a number with a real part and an imaginary part. The real part is any real number. The imaginary part is a real number multiplied by i. i represents -1. Since the coefficient of the imaginary part can be 0, all real numbers are also complex numbers.

More information

  • McAdams, David. Complex Number. All Math Words Encyclopedia. http://www.allmathwords.org/article.aspx?lang=en&id=Complex Number.
3+2i
-2.7+4i
e3.7i
Real Number A real number is a number that can be found on the real number line. All real numbers are also complex numbers. Stated mathematically: ℝ ⊂ ℂ.

More Information

  • McAdams, David. Real Number. All Math Words Encyclopedia. http://www.allmathwords.org/article.aspx?lang=en&id=Real%20Number.
4
3.74
e2.0
5
π
3.5193
Imaginary Number Double struck I An imaginary number is a complex number with no real part. All imaginary numbers are also complex numbers.

More Information

4i
3.74i
e3πi/2
-3
Rational Number A rational number is a real number that can be expressed as the ratio of two integers. A number with a repeating decimal is a rational number, as all repeating decimals can be expressed as a ratio of integers. All rational numbers are also real numbers. Stated mathematically: ℚ ⊂ ℝ.

More Information

  • McAdams, David. Rational Number. All Math Words Encyclopedia. http://www.allmathwords.org/article.aspx?lang=en&id=Rational Number.
4 Why?
3/7
9 Why?
3.5193 Why?
Irrational Number None An irrational number is a number that can not be expressed as the ratio of two integers. The set of rational numbers combined with the set of irrational numbers makes up the set of real numbers.

More Information

  • McAdams, David. Irrational Number. All Math Words Encyclopedia. http://www.allmathwords.org/article.aspx?lang=en&id=Irrational Number.
π
e
5
Integer An integer is a positive or negative whole number or zero. All integers are also rational numbers. This is because any integer a can be written a/1.

More Information

  • McAdams, David. Integer. All Math Words Encyclopedia. http://www.allmathwords.org/article.aspx?lang=en&id=Integer.
4
7
9 Why?
3.0 Why?
Natural Number A natural number is the set of positive integers: {1, 2, 3, ...}. These are also called counting numbers. All natural numbers are also integers.

More Information

  • McAdams, David. Natural Number. All Math Words Encyclopedia. http://www.allmathwords.org/article.aspx?lang=en&id=Natural%20Number.
4
6/2
Table 2: Types of numbers
Properties of Numbers
PropertyDescription
0 - addition byFor any real or complex number a, a + 0 = 0 + a = a. See also additive identity
0 - multiplication byFor any real or complex number a, a·0=0·a=0.
0 - division byDivision by zero is undefined.
1 - multiplication byFor any real or complex number a, 1·a=a·1=a.
Absolute valueThe absolute value of a number is the distance of a number from zero. If a≥0, |a|=a. If a<0, |a|=-a.
AdditionAddition is combining two numbers to form a sum. Addition of real and complex numbers is associative, commutative and closed.
Additive identityThe additive identity for real and complex numbers is 0 since, for any real or complex number a, a + 0 = 0 + a = a.
Additive inverseFor all real and complex numbers a, the additive inverse of a is -a.
Axiom of ArchimedesThere is always at least one more number. For every x∈ℜ, there exists a number n∈ℜ such that n>x.
Division by 0Division by zero is undefined.
EqualityEquality is a property that two numbers are either equal to each other or not equal to each other. Equality of numbers is symmetric, reflexive and transitive.
MultiplicationMultiplication is repeated addition. a·b = a+a+a+...+a b times. Multiplication of real and complex numbers is associative, commutative and closed.
Multiplicative identityThe multiplicative identity for real and complex numbers is 1 since, for any real or complex number a, a·1 = 1·a = a.
Multiplicative inverseFor all real and complex numbers a, the multiplicative inverse of a is 1/a.
Ruler PostulateThe Ruler Postulate defines the association of the real numbers with a number line.

Special numbers
NameDescription
Composite numberA composite number is an integer greater than 1 that has factors other than 1 and itself.
FactorialThe factorial of n is the product of all integers from 1 to n, inclusive.
Fibonacci numbersThe Fibonacci numbers are a sequence of numbers starting with 0, 1. After the first two numbers, each number in the sequence is calculated by adding the two previous numbers.
GoogolA googol is a very large number. 1 googol = 10100.
GoogolplexA googolplex is an extremely large number. 1 googolplex = 10googol.
Prime numberA prime number is an integer greater than 1 that is divisible only by 1 and itself.

More Information

  • McAdams, David. Complex Number. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Complex%20Number.
  • McAdams, David. Integer. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Integer.
  • McAdams, David. Prime number. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Prime%20Number.
  • McAdams, David. Real number. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Real%20Number.
  • McAdams, David. Real number. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Number%20Line.
  • Wilson, Robin. 4000 Years of Numbers. Gresham College. 2009-03-12. http://www.gresham.ac.uk/event.asp?PageId=45&EventId=622.

Cite this article as:


Number. 2010-08-05. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/n/number.html.

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Revision History


2010-08-05: Added properties of numbers, special numbers (McAdams, David.)
2009-12-19: Added "References" (McAdams, David.)
2008-12-02: Changed equations to images (McAdams, David.)
2008-08-26: In the first paragraph, organized representations of 3 into a table (McAdams, David.)
2008-06-08: Added types of numbers (McAdams, David.)
2008-06-07: Corrected spelling (McAdams, David.)
2008-05-02: Revised representation of irrational numbers (McAdams, David.)
2008-03-22: Changed Other Information to current standard (McAdams, David.)
2007-11-20: Initial version (McAdams, David.)

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