GED Mathematical Reasoning: Number Line And Signed Numbers II – Subtracting

Subtracting with signed numbers

  • Subtraction is the opposite of addition. Negatives are the opposite of positives. And all this leads to this very important fact: We can think of subtraction as “adding the opposite.” This means: Any subtraction problem can be changed to an addition problem as long as we change the number being subtracted to its opposite. After we change the subtraction sign to the addition operation, we can then use the steps for adding signed numbers to find the answer.
  • To add numbers with the same sign, add the number parts and use the common sign.
  • To add numbers with different signs, take the difference of the number parts and use the sign of the number having the larger number part.

 

Example 1

Subtract: 6 – 2

 

It probably didn’t take you long to determine the answer of 4. Visually, using a number line, this problem tells us to start at zero and move 6 units to the RIGHT – in the positive direction. Then, turn around and move two units to the LEFT– in the negative direction – because it’s as if the minus sign makes the 2 negative. We land on the answer of ‘positive 4.’

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Let’s solve it without using the number line. ‘6 minus 2’ can be thought of as: “six add the opposite of 2.” This means it can be rewritten as ‘6 plus negative 2.’ We keep the first number the same, change the subtraction sign to addition and change the number 2 to its opposite of -2.

6 – 2 = 6 + (-2)

Now we’re adding two numbers with different signs, so we’ll find the difference between the number parts, which is 4, and use the positive sign in our answer since the number with the larger number part is 6 and the 6 is positive.

6 – 2 = 6 + (-2) =+4 or just 4

 

Example 2

Subtract: 3 – 5

 

Let’s see what happens when we use a number line to find the answer.

Visually, using a number line, this problem tells us to start at zero and move 3 units to the RIGHT – in the positive direction. Then, we’ll turn around and move 5 units to the LEFT– in the negative direction – because it’s as if the minus sign makes the 5 negative. We land on the answer of ‘negative 2.’

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Let’s try it without using a number line.

‘3 minus 5’ can be thought of as: “3 add the opposite of 5.” This means it can be rewritten as ‘3 plus negative 5.’ We keep the first number the same, change the subtraction sign to addition and change the number 5 to its opposite of -5.

3 – 5 = 3 + (-5)

Now we’re adding two numbers with different signs, so we’ll find the difference between the number parts, which is 2, and use the negative sign in our answer since the number with the larger number part is 5 and the 5 is negative.

3 + (-5) = -2

And this makes sense if you think of the example in terms of money in a bank account. Start with a zero balance and imagine making a deposit $3. If you then withdraw $5, the account will have a negative $2 balance.

 

Example 3

Subtract: (-9) – (-11)

 

‘-9 minus -11’ can be thought of as: “-9 add the opposite of -11.”  We keep the first number the same, change the subtraction sign to addition and change the number -11 to its opposite of  +11.

(-9) – (-11) = (-9) + (+11)

To add these numbers with different signs, we find the difference between the number parts 9 and 11, which is 2, and use the positive sign in our answer since the number with the larger number part is 11 and the 11 is positive.

(-9) + (+11) = +2 or just 2

 

Example 4

Subtract: 18 – (+15)

 

‘18 minus positive 15’ can be thought of as: “18 add the opposite of +15.”  We keep the first number the same, change the subtraction sign to addition and change the number +15 to its opposite  -15.

18 – (+15) = 18 + (-15)

To add these numbers with different signs, we find the difference between the number parts 18 and 15, which is 3, and use the positive sign in our answer since the number with the larger number part is 18 and the 18 is positive.

18 + (-15) =+3 or just 3

 

Example 5

Subtract:  -34 – (-4)

 

‘-34 minus -4’ can be thought of as: “-34 add the opposite of -4.”  We keep the first number the same, change the subtraction sign to addition and change the number -4 to its opposite ‘positive 4’.

-34 – (-4) = -34 + (+4)

To add these numbers with different signs, we find the difference between the number parts 34 and 4, which is 30, and use the negative sign in our answer since the number with the larger number part is 34 and the 34 is negative.

-34 + (+4) = -30

 

Example 6

Subtract:  42 – (-16)

 

‘42 minus -16’ can be thought of as: “42 add the opposite of -16.” We keep the first number the same, change the subtraction sign to addition and change the number -16 to its opposite ‘positive 16’.

42 – (-16) = 42 + (+16)

Now we just have a simple addition problem with positive numbers. We’ll add 42 and 16, which is 58 and use the positive sign since both numbers are positive.

42 + (+16) = 58

 

Example 7

7 – (-2) + 10 – 3 – 16

 

A good first step when you encounter a problem involving the subtraction of signed numbers is to change all the subtraction signs to addition using the “add the opposite” strategy.

‘minus -2’ means “add the opposite of -2” so it becomes ‘plus positive 2’

’minus 3’ means “add the opposite of +3” so it becomes ’plus -3’

And ‘minus 16’ means “add the opposite of +16” so it becomes ‘plus -16’

7 - (-2) + 10 - 3 - 16 = 7 + (+2) + 10 + (-3) + (-16)

Now that we have changed all the subtraction signs to addition and are essentially adding a bunch of numbers, the order in which we add does not matter. Let’s add all the positive numbers, then add all the negative numbers. Finally, we’ll combine those two results for the final answer.

The three positive numbers are 7, 2 and 10. They add to be: 19.

The negative numbers are -3, and -16. They add to be: -19.

Since the numbers have different signs, we’ll find the difference of the number parts 19 and 19, which is zero. Since zero can be neither positive nor negative, the final answer is simply zero.

7 + (+2) + 10 + (-3) + (-16) = 19 + (-19) = 0

In fact, anytime you add a number and its opposite, the result will be zero.

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