Integers: Positive and Negative Whole Numbers

Rises and Falls

In everyday life, we most commonly use negative numbers when talking about temperature and money -- particularly profits and losses. The first measuring device we meet that includes negative numbers is a thermometer -- especially those of us who live in the Northern Hemisphere.

Let's look at a thermometer.

Let's investigate the last statement. We know that if the temperature is 3° and it drops 5°, the result will be – 2°. That's because (+ 3) + (– 5) = + 3 + (– 3 – 2) and 3 + (– 3 ) = 0, so we're left with – 2. Notice how a "drop" of 5° is indicated by the negative number -5. Now let's say the initial temperature is – 4°. If it rises 9°, the result will be 5°, because

(– 4) + (+ 9) = (– 4) + (4 + 5) = 5.

Once the column in the thermometer rises 4°, it will be at 0°. Then, to complete the rise of 9°, it will continue going up until it reaches 5°.

On the thermometer, a rise in temperature moves the indicator upwards. A rise is indicated by a positive number. A drop in temperature moves the indicator downward, so a drop in temperature is indicated by a negative number. The negative sign indicates a loss or drop.

adding a negative is the same as subtracting

On a vertical scale, like the thermometer, numbers below zero (0) are negative and those above zero are positive. A rise in temperature moves it upwards, a drop in temperature moves it downwards.

Example: If the temperature is –7° C and it drops 5° C, what is the new temperature?

Solution: Here, we have –7° + (– 5°) which takes us to –12°. It started at 7° below zero, then dropped (moved downwards) another 5° -- so now the temperature is 12° below zero.

Generally we'd write this as –7° – 5° = – 12°, leaving out the + sign and the brackets.

Gains and Losses

Say you're on Jeopardy and you have \$800. You then answer a \$2000 question incorrectly. Your score will then be – \$1200. Once they deduct \$800, you will be at \$0 -- then they deduct \$1200 more to finish taking off a total of \$2000.

\$800 + (– \$2000) = \$800 + (– \$800 – \$1200) = – \$1200.

To indicate this situation, we use a horizontal scale called a number line.

On a number line, numbers left of zero (0) are negative
and those right of zero are positive.

A gain moves the indicator towards the right,
a loss moves the indicator towards the left.

The question above could be written 5 – 3 – 4 = – 2 since adding a negative amount is exactly the same as subtracting a positive amount. We could also write the question above this way:

5 – (+ 3) – (+ 4) = – 2.

the result of 2 adjacent opposite signs is negative.

Because of this, when we remove brackets, we write a single negative sign for each pair of adjacent opposite signs. It makes no difference if the negative precedes or follows the positive, as long as the signs are adjacent.

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Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

1) For each question, draw a number line, indicate the moves and find the sum.

 a) 7 + (+ 2) + (–10) b) – 3 + (– 5) + (+ 12) c) –1 + (– 6) + (+ 9) d) 4 + (– 2) + (– 5)

2) Remove brackets, replace each pair of adjacent opposite signs with a negative, find the sum.

 a) 2 + (– 4) – (+ 10) b) 6 – (+ 3) + (– 7) c) 1 + (– 9) – (+ 6) d) 12 – (+ 8) + (–16)

Now continue with the lesson

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Subtracting Negative Numbers: Taking Away a Loss

When we subtract a negative amount, we have to add, because taking away a loss results in a gain. For instance 7 – (– 3) = 7 + 3 = 10. If we start at 7 and take away 3 steps left, we have to move 3 steps right. Think about it this way:

The first minus sign means the operation of subtraction -- take it away!

The second minus sign assigns a negative quality to the number or variable -- a loss or a debt.

When you take away a loss or a debt, there has to be a gain.

To take away 3 negative degrees, we must add 3 positive degrees.

To take away (1st minus) 7 steps left (2nd minus) -- we must go 7 steps right -- yes?

Say you owe me \$10.00 but you have no money in your pocket now to pay your debt.

You have -\$10.00 (negative ten dollars); you owe me \$10.00 and you have nothing -- right?

I'm feeling generous for some strange reason and I say "Forget about the \$10 you owe me."

Now, you have \$0 , which is \$10 more than you had before I took away a debt of \$10.

The 1st minus sign instructs us to take away (an operator) and

the 2nd minus sign is the negative quality of the \$10.00, because it is a debt.

So 0 – (–10) = 0 + 10 and 5 – (– 3) = 5 + 3 = 8.

and 3 – 2( 5) = 3 + 10 = 13

Subtracting a negative amount is the same as adding a positive amount.

The result of 2 adjacent identical signs is positive.

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Examples:

 a) 2 – (– 4) = 2 + 4 = 6 b) 6 – (– 3) = 6 + 3 = 9 c) (– 9) – (– 6) = – 9 + 6 = – 3

replace every pair of adjacent unlike signs with a minus ( – )
replace every pair of adjacent like signs with a plus ( + ).

Examples:

 a) 5 + (– 3) – (–7) =5 – 3 + 7 =2 + 7 = 9 b) (–4) – (–1) – (+ 9) =– 4 + 1 –9 =–3 – 9 = –12 c) 12 + (–7) –(–20) =12 – 7 + 20 =5 + 20 = 25 d) –2 – (– 5) – (– 4) =–2 + 5 + 4 =3 + 4 = 7

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Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

Practice Exercise 2: Subtracting Integers

1) Remove brackets, replace each pair of adjacent signs with a + or – , then find the sum.

 a) 32 + (– 14) – (– 10) b) 16 – (– 3) + (– 7) c) 1 – (– 9) + (+ 6) d) 12 – (+ 8) – (– 6) e) 2 – (+ 4) – (– 9) f) 18 + (+ 4) – (– 26) g) 17 – (– 34) – (– 5) h) 13 – (– 5) + (– 17)

2) On March 1st, Chucky had \$75 in his bank account. He deposited his pay check in the amount of \$450 and withdrew (took out) \$100 to cover some expenses. During the month of March, he wrote 3 checks in the amounts of \$34, \$112 and \$65. Using + for deposits and – for checks and withdrawals,
a) write a number sentence to show the activity in Chucky's account for the month of March.
b) How much money does he have left in the account on April 1st?

Solutions

1) For each question, draw a number line, indicate the moves and find the sum.

 a) 7 + (+ 2) + (– 10)start at 7, 2 steps right 10 steps left7 + 2 – 10 = –1 b) –3 + (– 5) + (+ 12)start at –3, 5 steps left 12 steps right– 3 – 5 + 12 = 4 c) –1 + (– 6) + (+ 9)start at –1, 6 steps left 9 steps right–1 – 6 + 9 = 2 d) 4 + (– 2) + (– 5)start at 4, 2 steps left, 5 steps left4 – 2 – 5 = – 3

Number Lines:

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2) Remove brackets, replace each pair of adjacent opposite signs with a negative, find the sum.

 a) 2 + (– 4) – (+ 10)2 – 4 – 10 = – 12 b) 6 – (+ 3) + (– 7)6 – 3 – 7 = – 4 c) 1 + (– 9) – (+ 6)1 – 9 – 6 = – 14 d) 12 – (+ 8) + (– 16)12 – 8 – 16 = – 12

Practice Exercise 2: Subtracting Integers

1) Remove brackets, replace each pair of adjacent signs with a + or – , then find the sum.

 a) 32 + (– 14) – (– 10)32 – 14 + 10 = 28 b) 16 – (– 3) + (– 7)16 + 3 – 7 = 12 c) 1 – (– 9) + (+ 6)1 + 9 + 6 = 16 d) 12 – (+ 8) – (– 6)12 – 8 + 6 = 10 e) 2 – (+ 4) – (– 9)2 – 4 + 9 = 7 f) 18 + (+ 4) – (– 26)18 + 4 + 26 = 48 g) 17 – (– 34) – (– 5)17 + 34 + 5 = 56 h) 13 – (– 5) + (– 17)13 + 5 – 17 = 1

2) a) \$75 + \$450 – \$100 – \$34 – \$112 – \$65 = \$214.00

b) Chucky has \$214.00 in his account on April 1st.

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