The numbers on this line are called INTEGERS. Every integer (except zero) consists of a sign and a magnitude. The sign indicates the direction from zero; a “-” indicates it is to the left of zero and a “+” indicates it is to the right of zero. Numbers with a “-“ in front are called “negative” numbers while numbers with a “+” in front are called “positive” numbers. If there is no sign then it is assumed that the number is positive.

The magnitude indicates the distance from zero. For example, -8 has a magnitude of 8 and +4 (or just 4) has a magnitude of 4.

What is the magnitude of -12?- 10
- 12
- 14
- -12

Is 8 positive or negative?

- Positive
- Negative
- Neither positive nor negative.

What is the magnitude of 87?

- 100
- 97
- 87

Zero is considered to have no sign and a magnitude of 0. Is zero positive or negative?

- Positive
- Negative
- Neither positive nor negative

When adding and subtracting integers, you are really moving to the left and right along the number line. For example, if you add 5 to a number, you move to the right 5 places.

What is 2 + 5?- -3
- 2
- 5
- 7

When you subtract 5 from a number, you move 5 places to the left.

What is 2 - 5? (Hint: The answer will be a negative number.)- -3
- 2
- 5
- 7

What is -2 + 3? (Hint: Start at -2 and move 3 places to the right.)

- 1
- 5
- -5

What is -2 - 3?

(Hint: Start at -2 and move 3 places to the left.)- 1
- -5
- -7

What is 5 - 8?

- 13
- 3
- -3
- -13

What is 10 - 3?

- 13
- 10
- 7

What is -3 + 4?

- 7
- -1
- 1
- -7

What is -7 + 7?

- 14
- -14
- 0

When adding a negative # to an integer you end up moving to the left. This is because when you add, you move in the direction associated with the number you are adding. Negative #’s are to the left of zero, so when you add a negative number you move to the left. For example, if you add -2 to 5 you get 5 + (-2) = 3. That is, you move 2 places to the left of 5.

What is 7 + (-3)?- 10
- 7
- 4
- 3

What is 10 + (-2)?

- 12
- 8
- 2
- 0

Adding a negative number is actually the same as subtracting.

For example, 10 + (-2) is the same as 10 - 2 and 7 + (-3) is the same as 7 - 3.

What is 13 + (-5)?- 18
- 15
- 8
- 5

What is 48 + (-23)?

- 71
- 61
- 55
- 25

What is 235 + (-58)?

- 177
- 187
- 293
- 327

When you subtract a negative number you move to the right on the number line. Can you think why? When you subtract you are going in the opposite direction of the number you are subtracting. For example, when you subtract -2 from 5 you move two places to the right of 5, so 5 - (-2) = 7.

What is 4 - (-3)?- 1
- 4
- 7
- 10

What is 3 - (-6)?

- -3
- 3
- 6
- 9

What is (-2) - (-5)?

- -7
- -3
- 3
- 7

What is (-7) - (-3)?

- -10
- -4
- 0
- 4

So subtracting a negative number is equivalent to adding. That is, 5 - (-2) = 5 + 2 = 7.

What is (-12) - (-5)?- -17
- -7
- 7
- 17

What is 57 - (-6)?

- 63
- 51
- 49
- 44

What is (-28) - (-14)?

- 14
- 0
- -14

What is 108 - (-226)?

- 334
- 118
- -118

For multiplying integers, there is a simple rule you need to remember. A positive number multiplied by a positive number results in a positive product. A negative times a negative results in a positive. A positive times a negative (or a negative times a positive) results in a negative product.

What sign is the result when you multiply a negative number by a negative number?- Positive
- Negative
- Neither

After you have figured out the sign of the result, you can find the magnitude simply by multiplying the two magnitudes. Put the sign in front and you’re done. For example,

2 X 2 = 4

2 X (-2) = -4

(-2) X 2 = -4

(-2) X (-2) = 4

What is 3 X (-3)?- 9
- -9
- 18
- -18

What is -3 X (-5)?

- 15
- 8
- -2
- -15

What is 5 X 6?

- 15
- -15
- 30
- 11

What is -6 X (-3)?

- 18
- -18
- -9
- 6

What is (-7) X 7?

- 14
- -14
- 49
- -49

What is 12 X (-12)

- 144
- 0
- -144
- 24

For dividing, use the same rule for finding the sign and just divide the unsigned numbers. For example,

4 ÷ (-2) = -2

(-4) ÷ 2 = -2

4 ÷ 2 = 2

(-4) ÷ (-2) = 2

What is 6 ÷ (-2)?- -3
- 3
- 2
- 1

What is (-8) ÷ (-2)?

- 10
- 4
- -4
- -10

What is 25 ÷ 5?

- 5
- 4
- -5
- -125

What is 25 ÷ (-5)?

- 5
- 4
- -5
- -125

What is (-5) ÷ (-5)

- 25
- -10
- -25
- 1

What is 100 ÷ (-4)

- 104
- 25
- 10
- -25

Evaluate: 5(-4) + 2

- 18
- -22
- 22
- -18

Evaluate: 3 - (-3)(4)

- 15
- 12
- -6
- 9

Evaluate: 4(-2) + 3(-4)

- 20
- 4
- -4
- -20

Evaluate: 7 + 3(-6)

- 18
- 11
- -18
- -11

Evaluate: -2(3) - (-4)(-3)

- 20
- 18
- 4
- -18

Do you remember how to solve for 'x' in an equation? These questions are similar to the ones in the Expressions exercise, but now you will run into negative numbers here are there.

Solve for x:

x + 4 = 2- x = 2
- x = -2
- x = 6
- x = -6

Solve for x:

x - 2 = -6- 10
- -10
- 4
- -4

Solve for x:

2x + 4 = 2- 3
- 2
- 1
- -1

Solve for x:

4x - 2 = 6x + 8- -5
- -6
- -8
- -10

Solve for x:

-7x = 49- -7
- 7
- 14
- 21

Exponents can also be negative. To understand negative exponents, think about what happens with the problem 2

^{2}÷ 2^{3}.

If we use the exponent law for division, we get 2^{-1}, but what does this mean?

A division also represents a fraction so the problem can also be written like this: 2^{2}/ 2^{3}, which can be expanded to (2X2) / (2X2X2), which after canceling becomes 1/2. So, 2^{-1}= 1/2.

Similarly, 2^{-2}= 1/2^{2}, 2^{-3}= 1/2^{3}, 2^{-4}= 1/2^{4}and so on.

In general, a negative exponent can be evaluated by changing it to a positive exponent and putting 1 over the whole thing (so we get a fraction). This makes sense because any number to the exponent zero is 1. So we can show that 2^{-4}= 1/2^{4}by using the exponent rule for division (if the bases are the same then subtract the exponents):

1/2^{4}= 2^{0}/ 2^{4}= 2^{0 - 4}= 2^{-4}

Evaluate 2^{-5}. (Hint: Answer will be a fraction).- 1/2
- 1/4
- 1/16
- 1/32

Evaluate: 3

^{-1}- 3
- 1
- 1/3

Evaluate: 3

^{-2}- -9
- -6
- 1/3
- 1/9

Evaluate: 10

^{-1}- 10
- -10
- 1/10
- -1/10

Evaluate: 4

^{-3}- 1/12
- -1/12
- 1/64
- 1/256