Integers, Rational and Real Numbers

Mr. Donnelly's Math Page

Integers, Rational and Real Numbers

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Signed numbers are also called integers (Set Z) which includes all negative numbers, all positive numbers and zero (0 is not positive or negative.)

We will assume a number without a sign (except zero) to be positive.

Multiplication and Division:

When we multiply and divide signed numbers we will perform the operations as always but remember two rules:

1. If both numbers are the same sign the answer is positive. (A - times a - is a +.)

2. If the signs are different the answer is a negative

Addition and Subtraction:

These can be a bit harder but a good trick is to apply the same rules as above and make problems simpiler.

Look at the operation and the sign of the second number, apply the rules above and covert the two signs to one.

Example 1:

                  7 +(-8)=

The two signs are different so we can combine them as a - and get a simpler problem

                 7 - 8 = -1

Example 2:

                -11 - (-15)=

The two negatives become a positive

                -11 + 15=

Remember addition is commutative which means order does not matter so...

                -11+ 15= 15 +(-11)

Now we have +- which is - so...

                 15-11 = 4

Hopefully this makes things a little easier.

Number Systems

Everyone knows what the counting numbers are:

1,2,3,4,5...

Notice no zero! These are also known as Natural Numbers. When we include zero we say we have the whole numbers.

The natural numbers are what we call "closed" under addition and multiplication. This means if we take any two natural numbers and add them or multiply them the result is also a natural number.

This is not true for subtraction; 3 -5 doesn't yield a natural answer. For this we need negative numbers. The set of all the positive whole numbers, and all negative whole numbers is called the integers.

Note: Integers do not include fractions or decimals.

Integers are closed for subtraction but not for division. Sure some division problems have integer quotients (example -12/4= -3) but not all. 4/-12= -1/3 or .3333...

All of these answers give us a set we call the Rational Numbers (as in ratio; in fact we call this set Q, Q as in quotient.)

So were done right, we've used all 4 operations; sorry not quite.

What about square roots? Perfect squares like 1,4,9,16,25... have rational (in fact integer) roots but numbers like the square root of 10 (or the square root of any number except the perfect squares) cannot be shown as ratios or terminating decimals. They are irrational. Pi is also an irrational number.

Hint: If a decimal repeats or ends it is rational; non-terminating, non repeating decimals are irrational numbers.

Finally, If we take all the rational numbers and all the irrational numbers we have what we call the Real Numbers. Simply, any number that can be represented on the number line (sometimes called the real line) is a real number.

What is not a real number? The square root of -9...

no it's not -3, -3 * -3 = +9

Negative numbers do not have real roots, more on this later...