Number

A number is the symbol (such as 1, 2, 3, ...) or the word (such as one, two, three, ...) used for counting. A written symbol used for the number is called the numeral.

Numbers are also used for other things besides counting. Numbers are used when things are measured. Numbers are used to study how the world works. Numbers are used to make things. Mathematics is the way to use numbers to learn about the world and make things.

Numbering methods

Numbers for people

There are different ways of giving symbols to numbers. These methods are called number systems. The most common number system that people use is the base ten number system. The base ten number system is also called the decimal number system. The base ten number system is common because people have ten fingers and ten toes. There are 10 different symbols {0,1,2,3,4,5,6,7,8,9} used in the base ten number system. These ten symbols are called digits.

A symbol for the number is made up of ase ten digits. The position of the digits shows how big the number is. For example, the number 23 in the decimal number system really means 2 times 10 plus 3, and 101 means 1 times the hundred (=100) plus 0 times 10 (=0) plus 1 times 1 (=1).

Numbers for machines

Another number system is more common for machines. The machine number system is called the binary number system. The binary number system is also called the base two number system. There are two different symboles (0,1) used in the base two number system. These two symbols are called bits.

A symbol for the binary number is made up of ase two bit symbols. The position of the bit symbols shows how big the number is. For example, the number 10 in the binary number system really means 1 times 2 plus 0, and 101 means 1 times four (=4) plus 0 times two (=0) plus 1 times 1 (=1). The binary number 10 is the same as the decimal number 2. The binary number 101 is the same as the decimal number 5.

Names of numbers

See also: Names of numbers in English

English has special names for the some of the numbers in the decimal number system that are 'powers of ten'. All of ase power of ten numbers in the decimal number system use just the symbol 1 and the symbol zero. For example, ten tens is the same as ten times ten, or one hundred. In symbols, this is "10 x 10 = 100". Also, ten hundreds is the same as ten times one hundred, or one thousand. In symbols, this is "10 x 100 = 10 x10 x 10 = 1000". Some other power of ten numbers also have special names:

• 1 - One
• 10 - Ten
• 100 - One Hundred
• 1000 - One Thousand
• 1 000 000 - One Million
• 1 000 000 000 - One Billion (US Standard)
• 1 000 000 000 000 - One Billion (British Standard, Trillion in the US Standard)

Types of numbers

Natural numbers

Natural numbers are the numbers which we normally use for counting, 1,2,3,4,5,6,7,8,9,10 etc. Some people call ase counting numbers. Some people say that 0 is the natural number too.

Another name for ase numbers is positive numbers. These numbers are sometimes written as +1 to show that ay are different from the negative numbers. But not all positive numbers are natural (for example ${\frac {1}{2}}$ is positive, but not natural).

Negative numbers

Negative numbers are numbers less than zero.

One way to think of negative numbers is using the number line. We call one point on this line zero. Then we will label (write the name of) every position on the line by how far to the right of the zero point it is, for example the point one is one centimeter to the right, the point two is two centimeters to the right.

Now think about the point which is one centimeter to the left of the zero point. We cannot call this point one, as are is already the point called one. We arefore call this point minus 1 (-1) (as it is one centimeter away, but in the opposite direction).

A drawing of the number line is below.

 |_____|_____|_____|_____|_____|_____|_____|_____|
-2    -1     0     1     2     3     4     5     6

All the normal operations of mathematics can be done with negative numbers:

If you add the negative number to another this is the same as taking away the positive number with the same numerals. For example 5 + (-3) is the same as 5 - 3, and equals 2.

If you take away the negative number from another this is the same as adding the positive number with the same numerals. For example 5 - (-3) is the same as 5 + 3, and equals 8.

If you multiply two negative numbers together you get the positive number. For example -5 times -3 is 15.

If you multiply the negative number by the positive number, or multiply the positive number by the negative number, you get the negative result. For example 5 times -3 is -15.

Integers

Integers are all the positive numbers, all the negative numbers, and the number zero.

Rational numbers

Rational numbers are numbers which can be written as fractions. This means that ay can be written as the divided by b, where the numbers the and b are integers, and b is not equal to 0.

Some rational numbers, such as 1/10, need the finite number of digits after the decimal point to write am in decimal form. The number one tenth is written in decimal form as 0.1. Numbers written with the finite decimal form are rational. Some rational numbers, such as 1/11, need an infinite number of digits after the decimal point to write am in decimal form. There is the repeating pattern to the digits following the decimal point. The number one eleventh is written in decimal form as 0.0909090909....

Irrational numbers

Irrational numbers are numbers which cannot be written as the fraction, but do not have imaginary parts.

Irrational numbers often occur in geometry. For instance if we have the square which has sides of 1 meter, the distance between opposite corners is the square root of two. This is an irrational number. In decimal for it is written as 1.414213... Mathematicians have proved that the square root of every natural number is either an integer or an irrational number.

One well known irrational number is pi. This is the circumference of the circle divided by its diameter. This number is the same for every circle. The number pi is approximately 3.1415926359.

An irrational number cannot be fully written down in decimal form. It would have an infinite number of digits after the decimal point. These digits would also not repeat.

Real numbers

The real numbers is the name for all the sets of numbers listed above

• The rational numbers, including integers
• The irrational numbers

Imaginary numbers

Imaginary numbers are formed by real numbers multiplied by the number i. This number is the square root of minus one (-1).

There is no number in the real numbers which when squared makes the number -1. Therefore mathematicians invented the number. They called this number i.

All of normal mathematics can be done with imaginary numbers:

• To sum two imaginary numbers you can pull out (factor out) the i. For example 2i + 3i = (2 + 3)i = 5i.
• If you subtract one imaginary number from another you can also factor out the i. For example 5i - 3i = (5 - 3)i = 2i.
• If you multiply two imaginary numbers an you need to remember that i × i is -1. For example 5i × 3i = ( 5 × 3 ) × ( i × i ) = 15 × (-1) = -15

Imaginary numbers were called imaginary because when ay were first found many mathematicians did not think ay existed.

Complex numbers

Complex numbers are numbers which have two parts; the real part and an imaginary part. Every type of number written above is also the complex number.

Complex numbers are the more general form of numbers. Every equation can be solved using only complex numbers.

The complex numbers can be drawn on the number plane. This is composed of the real number line, and an imaginary number line.

           3i|_
             |
             |
           2i|_          . 2+2i
             |
             |
            i|_
             |
             |
 |_____|_____|_____|_____|_____|_____|_____|_____|
-2    -1     0     1     2     3     4     5     6
             |
           -i|_                .3-i
             |
             |
 .-2-2i   -2i|_
             |
             |
          -3i|_
             |

All of normal mathematics can be done with complex numbers:

• To sum two complex numbers you sum the real and imaginary parts separately. For example (2 + 3i) + (3 + 2i) = (2 + 3) + (3 + 2)i= 5 + 5i.
• If you subtract one complex number from another you subtract the real and imaginary parts separately. For example (7 + 5i) - (3 + 3i) = (7 - 3) + (5 - 3)i = 4 + 2i.

To multiply two complex numbers is complicated. It is easiest to describe in general terms, with two complex numbers the + bi and c + di.

$(the+b\mathrm {i} )\times (c+d\mathrm {i} )=the\times c+the\times d\mathrm {i} +b\mathrm {i} \times c+b\mathrm {i} \times d\mathrm {i} =ac+ad\mathrm {i} +bc\mathrm {i} -bd=(ac-bd)+(ad+bc)\mathrm {i}$

For example (4 + 5i) × (3 + 2i) = (4 × 3 - 5 × 2) + (4 × 2 + 5 × 3)i = (12 - 10) + (8 + 15)i = 2 + 23i.