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

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.

.

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 1: Adding Integers**

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) |

( *solutions* )

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**

.

**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

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.**

.

**Examples:**

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

To add and subtract integers:

replace every pair of **adjacent **__unlike__** **** signs** with a

replace every pair of

**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 |

.

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) |

( *solutions* )

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**

**Practice Exercise 1: Adding Integers**

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

a) 7 + (+ 2) + (– 10) start at 7, 7 + 2 – 10 = |
b) –3 + (– 5) + (+ 12) start at –3, – 3 – 5 + 12 = |
c) –1 + (– 6) + (+ 9) start at –1, –1 – 6 + 9 = |
d) 4 + (– 2) + (– 5) start at 4, 4 – 2 – 5 = |

** Number Lines**:

............

............

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

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

**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 = |
b) 16 – (– 3) + (– 7) 16 + 3 – 7 = |
c) 1 – (– 9) + (+ 6) 1 + 9 + 6 = |
d) 12 – (+ 8) – (– 6) 12 – 8 + 6 = |

e) 2 – (+ 4) – (– 9) 2 – 4 + 9 = |
f) 18 + (+ 4) – (– 26) 18 + 4 + 26 = |
g) 17 – (– 34) – (– 5) 17 + 34 + 5 = |
h) 13 – (– 5) + (– 17) 13 + 5 – 17 = |

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

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

*(all content of the MathRoom** © Tammy the Tutor; 2004 - ).*