Operations on Real Numbers (positive and negative)

Real numbers include numbers on both sides of the number line. We need to be able to perform

arithmetic operations ( +, -, x, ÷) on all real numbers, whether they are positive or negative.

Positive and Negative Numbers

Remember that on the number line, values increase in positive value when we move right, and numbers increase in negative value when we move left.

If we think of positive and negative numbers in terms of money, we can consider positive value as assets and negative value as debt. When we accumulate more assets we have more money, but accumulation of debt sends us further into the negative direction.

-3 -2 -1 0 1 2 3

We start at the point representing 3 and move 1.5 units to the right

Add a Positive Amount

So, if we are adding a positive amount to a number, we move to the right along the number line.

We start at the point -5 and move 2 units to the right. We are still in debt, but moving in the right direction.

3 + 1.5 = 4.5

-5 + 2 = -3

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

We start at the point 5.5 and move 4 units to the left

Add a Negative Amount

If we add a negative amount to a number, we move that number of units to the left.

We start at the point -2 and move 4 units to the left

5.5 + (-4) = 1.5

-2 + (-4) = -6

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

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

Rules for Addition of Real Numbers

• We can make some general rules so we don’t have to draw a number line for every problem:

• When adding numbers with like signs, we add their absolute values and use the sign of the original addends.

2.5 + 3 = |2.5| + |3| = 5.5-1 + (-4) = - (|-1| + |-4|) = - (1 + 4) = -5

Rules for Addition of Real Numbers

• When adding numbers with different signs, find the difference of their absolute values and use the sign of the addend with the greater absolute value.

-5 + 6 = + (|6| - |-5|) = + (6 – 5) = 1

37 + (-42) = - (|-42|- |37|) = - (42 – 37) = -5

Subtraction

Subtraction is the opposite of addition, so we can restate any subtraction problem as adding the inverse (additive inverse) of the number. This is helpful when we are working with negative and positive numbers.

5 – 2 = 5 + (-2) = 3

-1.25 – 2.5 = -1.25 + (-2.5) = -3.75

-1 – (-1.3) = -1 + 1.3 = 0.3

Multiplication

• We know that when we multiply two positive numbers we get a positive result.

2 • 4 = 8

• But what if one or both of the numbers are negative?

However, if that deduction were reversed 4 times, then you would be ahead $8.

(-2) • 4 = -8 Think in terms of money…. If $2 is deducted from your bank account 4 times, you are down $8.

(-2) • (-4) = 8

Division

Here is an example of division involving negative numbers

• Let’s say you have a debt of $30, represented by -30 • Now, two good friends come along and offer to help

pay the debt and split it 3 ways (-30) ÷3 = -10 You are still in debt, but only $10 now.

• Now, if the debt is reversed as well as divided, the problem looks like this:

(-30) ÷ (-3) = 10 The result is a credit to each rather than debt.

Generalization

We can make some general rules for multiplication and division of real numbers.

• If the signs are the same (both positive or both negative) we get a positive result.

2 • 2 = 4 (-2) • (-2) = 4 4 ÷ 2 = 2 (-4) ÷ (-2) = 2• If the signs are different, then the result is negative.

2 • (-2) = -4 (-2) • 2 = -44 ÷ (-2) = -2 (-4) ÷ 2 = -2

Multiple factors

Note that if we have an odd number of negatives, the result will be negative and if we have an even number ( or zero) negatives, the result will be positive

What if we have more than two factors ?

(-2) • (-3) • ( ) • (-5) • (6)

6

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Multiplying the first two negative numbers results in a positive 6.

Now we have 2 positives that also result in a positive value.

The next two result in negative

Final answer is negative

Impact on Exponents

What happens when we raise a negative number to some power?

= (-2) • (-2) • (-2) • (-2) • (-2) = -32 we have an odd number of negative factors = (-2) • (-2) • (-2) • (-2) • (-2) • (-2) = 64

we have an even number of negative factors.

To generalize: When raising a negative number to a power, if the exponent is odd, the result will be negative; if the exponent is even, the result will be positive.

Practice

Now, let’s combine what we know about Order of Operations with what we know about operations on positive and negative numbers.

First we perform operations within Parentheses

Next, resolve the Exponents

Multiply and Divide from left to right

Add and Subtract from left to right

-(64) – 4(-27) ÷ 9-64 + 108 ÷ 9-64 + 12 -52

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