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Home , Irrational number, Negative number, Rational number

The Evolution of

Counting Numbers

Counting Numbers: {1, 2, 3, …}

We use numbers to count: 1, 2, 3, 4, etc

 You can have "3 friends"  A can have "6 cows"

Whole Numbers

Whole numbers are the counting numbers plus zero.

Whole Numbers: {0, 1, 2, 3, …}

Negative Numbers

We can count forward: 1, 2, 3, 4, ...... but what if we count backward: 3, 2, 1, 0, ... what happens next?

The answer is: negative numbers: {…, -3, -2, -1}

A negative is any number less than zero.

Integers

If we include the negative numbers with the whole numbers, we have a new of numbers that are called : {…, -3, -2, -1, 0, 1, 2, 3, …}

The Integers include zero, the counting numbers, and the negative counting numbers, to make a that stretch in either direction indefinitely.

Rational Numbers

A is a number that can be written as a simple (i.e. as a ).

 2.5 is rational, because it can be written as the ratio 5/2  7 is rational, because it can be written as the ratio 7/1  0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3

More formally we say: A rational number is a number that can be written in the form p/q where p and q are integers and q is not equal to zero.

Example: If p is 3 and q is 2, then: p/q = 3/2 = 1.5 is a rational number

Rational Numbers include:

 all integers  all

Irrational Numbers

An is a number that cannot be written as a simple fraction. Another clue is that the goes on forever without repeating.

π () is an irrational number.  π = 3.1415926535897932384626433832795...  The decimal never repeats.  You cannot write a simple fraction that equals Pi. 22  The of /7 = 3.1428571428571... is close but not exact.

The root of 2 (√ퟐ ) is another example of an irrational number.

 √ퟐ = 1.4142135623730950...  The decimal goes on forever without repeating.  It cannot be written as a simple fraction.

Irrational numbers are useful. You need them to:

 find the diagonal distance across some  to work out calculations with

Real Numbers

Real Numbers include:  rational numbers, and  irrational numbers

A can be thought of as any point anywhere on the :

Imaginary Numbers

An , when squared, gives a negative result:

Think about this: if you multiply any number by itself you never get a negative result:

 2×2 = 4

 (-2)×(-2) = 4 because a negative times a negative gives a positive

So what number, when multiplied by itself, would result in -1? Mathematicians came up with such a number and called it i for imaginary:

ퟐ i x i = 풊 = -1

By taking the of both sides we get this:

ퟐ √풊 = √−ퟏ so i = √−ퟏ

In words, i is equal to the square root of -1.

We use i to answer questions like:

What is the square root of -9 ? Answer: √−9 = √(9) 푥 (−1) = √9 × √−1 = 3 × √−1 = 3i

There are many applications for Imaginary Numbers in the fields of electricity and electronics.

Complex Numbers

If you put a Real Number and an Imaginary Number together you get a new type of number called a . Here are some examples:

 3 + 2i  27.2 - 11.05i

A Real Number is a Complex Number with an imaginary part of 0:

 4 is a Complex Number because it is 4 + 0i

An Imaginary Number is a Complex Number with a real part of 0:

 7i is a Complex Number because it is 0 + 7i

Summary

Type of Number Quick Description

Counting Numbers {1, 2, 3, ...}

Whole Numbers {0, 1, 2, 3, ...}

Integers {..., -3, -2, -1, 0, 1, 2, 3, ...}

Rational Numbers Can be written as p/q : where p and q are integers, q is not zero

Irrational Numbers Cannot be written as a fraction, has a nonrepeating decimal

Real Numbers Rationals and Irrationals

Imaginary Numbers i , Squaring them gives a negative Real Number

Complex Numbers Combinations of Real and Imaginary Numbers

Pierce, Rod. "The Evolution of Numbers" Math Is Fun. Ed. Rod Pierce. 20 Nov 2014. 11 Jun 2015