Number System

Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that you may be better able to discover with accuracy the peculiar bent of the genius of each. ~ Plato

Numbers come from several different families. The simplest are the NATURAL NUMBERS, made up of normal counting numbers, 1, 2, 3, … Numbers

What happens when we have no more? ZERO ! These are called the WHOLE NUMBERS So now we have 0, 1, 2, 3, etc. but we are not yet quite complete. Numbers

If we count down from, say, three, we get Numbers Now what?

We often think of these numbers as arranged along a line: … … This line goes off as far as we like (to infinity) in either direction. We call all the numbers on this line the INTEGERS Numbers

There are lots of numbers which occur in between the integers – all the fractions, for example ½, -¾, 0.317, 2 1/3 etc. All the fractions and integers together are called RATIONAL NUMBERS, because they can all be written as ratios of whole numbers. Ratio is just an old word for fraction. Numbers

There are other numbers which cannot be represented by a fraction (unless we use an infinite number of decimal places). These are called IRRATIONAL NUMBERS and some you will be familiar with: √2 = … √3 = 1.732… π = … Numbers (This is special type of irrational, called a transcendental number)

All of these groups: Naturals N Wholes W Integers Z Rationals Q Irrationals I when added together make up the REAL NUMBERS Numbers

There is one final class of numbers whose members are not all in the REAL group and these are the COMPLEX NUMBERS which include things like √-1 or the square root of any other negative number. Numbers

You may not like complex numbers to start with, but, like the real numbers, they are extremely useful in calculations. We would probably have no electricity or certainly no electronic gadgets (cellphones, computers etc.) if people did not use complex numbers in their design. Numbers

Number Line We can picture integers as equally spaced points on a line called the number line. A whole number is graphed by placing a dot on the number line. The graph of 4 is shown

Comparing Numbers For any two numbers graphed on a number line, the number to the right is the greater number, and the number to the left is the smaller number. 2 is to the left of 5, so 2 is less than 5 5 is to the right of 2, so 5 is greater than

Comparing Numbers 2 is less than 5 is written as 2 < 5 5 is greater than 2 is written as 5 > 2 2 is to the left of 5, so 2 is less than 5 5 is to the right of 2, so 5 is greater than

One way to remember the meaning of the inequality symbols is to think of them as arrowheads “pointing” toward the smaller number. For example, 2 2 are both true statements. Helpful Hint

The integer –5 is to the left of –3, so –5 is less than –3 Since –3 is to the right of –5, –3 is greater than – –1–2–3–4–5–6 Comparing Numbers -5 < > -5

The absolute value of a number is the number’s distance from 0 on the number line. The symbol for absolute value is | |. is 2 because 2 is 2 units from –1–1–2–2–3–3–4–4–5–5–6–6 is 2 because –2 is 2 units from –1–1–2–2–3–3–4–4–5–5–6–6 Absolute Value

Since the absolute value of a number is that number’s distance from 0, the absolute value of a number is always 0 or positive. It is never negative. zero a positive number Helpful Hint

Two numbers that are the same distance from 0 on the number line but are on the opposite sides of 0 are called opposites. 5 units 5 and –5 are opposites –1–2–3–4–5–6 Opposite Numbers

5 is the opposite of –5 and –5 is the opposite of 5. The opposite of 4 is – 4 written as –(4) = –4 The opposite of – 4 is 4 written as –(– 4) = 4 –(–4) = 4 If a is a number, then –(– a) = a. Opposite Numbers

Remember that 0 is neither positive nor negative. Therefore, the opposite of 0 is 0. Opposites