The Evolution of Numbers Counting Numbers Counting Numbers: {1, 2, 3, …} We use numbers to count: 1, 2, 3, 4, etc You can have "3 friends" A field 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 number is any number less than zero. Integers If we include the negative numbers with the whole numbers, we have a new set of numbers that are called integers: {…, -3, -2, -1, 0, 1, 2, 3, …} The Integers include zero, the counting numbers, and the negative counting numbers, to make a list of numbers that stretch in either direction indefinitely. Rational Numbers A rational number is a number that can be written as a simple fraction (i.e. as a ratio). 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 fractions Irrational Numbers An irrational number is a number that cannot be written as a simple fraction. Another clue is that the decimal goes on forever without repeating. π (Pi) is an irrational number. π = 3.1415926535897932384626433832795... The decimal never repeats. You cannot write a simple fraction that equals Pi. 22 The approximation of /7 = 3.1428571428571... is close but not exact. The square 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 squares to work out calculations with circles Real Numbers Real Numbers include: rational numbers, and irrational numbers A Real Number can be thought of as any point anywhere on the number line: Imaginary Numbers An Imaginary Number, 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 square root 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 Complex Number. 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 <http://www.mathsisfun.com/numbers/evolution-of-numbers.html> .