2.2Addition and Subtraction of Integers

With negatives in play the number line runs both ways, so you can move left past $0$ .

A temperature of

-5°

C falling by

9°

C lands at

-14°

C. Moving left past

0

is the whole point.

Adding two negatives

\displaystyle (-a) + (-b) = -(a + b)

Adding a negative to a positive

\displaystyle a + (-b) = a - b

Subtracting a negative

\displaystyle a - (-b) = a + b

Temperature drop

At noon it was $\displaystyle -5\,^{\circ}\text{C}$ . By midnight it had fallen a further $\displaystyle 9\,^{\circ}\text{C}$ . Find the midnight temperature.

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Falling

9\,^{\circ}\text{C}

means subtracting:

(-5) + (-9)

Both moves go the same way down, so the falls add:

5 + 9 = 14

below zero.

Result is negative:

(-5) + (-9) = -14

The midnight temperature was

-14\,^{\circ}\text{C}

Altitude calculation

A submarine at $\displaystyle -120$ m rises $\displaystyle 45$ m, then dives $\displaystyle 30$ m. Find its final position relative to sea level.

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Start:

-120

Rise

45

-120 + 45 = -75

Dive

30

-75 + (-30) = -105

Final position:

-105

m, that is a depth of

105

m below sea level.

Paying off an overdraft

Ali's bank balance is $\displaystyle -\$85$ (overdrawn). He deposits $\displaystyle \$120$ . What is his new balance?

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(-85) + 120 = 120 - 85

= 35

His new balance is

\$35

Finding the starting temperature (reverse problem)

A temperature rose by $\displaystyle 11\,^{\circ}\text{C}$ to reach $\displaystyle -3\,^{\circ}\text{C}$ . Find where it started.

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A rise of

11

took it to

-3

, so go back down

11

from

-3

-3 - 11 = -14

It started at

-14\,^{\circ}\text{C}

Check:

-14 + 11 = -3

Now you try

Evaluate $7 - (-7)$ .

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\displaystyle 7 - (-7)

\displaystyle = 7 + 7 \quad \scriptsize\textit{subtracting a negative = adding}

\displaystyle = 14

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