Integers & Directed Numbers

On a number line, numbers further to the left are smaller. $-5$ is further left than $-2$, so $-5$ is smaller. (With negative numbers, the one with the larger digit is actually the smaller value.)

Start at $-3$ and move 8 to the right: $-3 + 8 = 5$. (The positive 8 outweighs the negative 3.)

Adding a negative is the same as subtracting: $4 + (-7) = 4 - 7 = -3$. (The negative 7 outweighs the positive 4.)

Subtracting moves further to the left: $-2 - 5 = -7$. (Two "downward" moves add together.)

Subtracting a negative is the same as adding: $-6 - (-4) = -6 + 4 = -2$. (Two minus signs together become a plus.)

A negative times a positive is negative: $-3 \times 4 = -12$.

A negative times a negative is positive: $-5 \times (-3) = 15$.

A negative divided by a positive is negative: $\dfrac{-20}{4} = -5$.

A negative divided by a negative is positive: $\dfrac{-18}{-6} = 3$.

The most negative number is smallest, so the order is $-7,\ -3,\ 0,\ 2,\ 5$. ($-7$ is smaller than $-3$ even though 7 is a bigger digit.)

Do multiplication before addition: $3 \times 4 = 12$, then $-2 + 12 = 10$. (Working left to right without the order of operations gives the wrong 4.)

Do the brackets first: $-2 + 3 = 1$, then $1 \times 4 = 4$. (The brackets change the answer compared with the previous question.)

The brackets mean the whole $-3$ is squared: $(-3)^2 = (-3)\times(-3) = 9$ (negative times negative is positive).

Without brackets, only the 3 is squared: $3^2 = 9$, then apply the negative to get $-9$. (Compare with $(-3)^2 = 9$: the brackets are the trap.)

Rising means adding: $-4 + 9 = 5$, so the new temperature is 5°C. (Falling would subtract.)

The fall is the difference: $3 - (-8) = 3 + 8 = 11$ degrees. (Count the steps: from 3 down to 0 is 3, then 0 down to $-8$ is 8, giving 11.)

Descending makes the depth more negative: $-50 + (-30) = -80$, so the new depth is $-80$ m.

A deposit adds to the balance: $-25 + 40 = 15$, so the new balance is $15.

Multiplying two negatives always gives a positive, for example $(-2)\times(-3) = 6$. (A negative times a positive, however, is negative.)

The student treated $-3 - 5$ as $-3 + 5$. But subtracting 5 from $-3$ moves further left on the number line: $-3 - 5 = -8$.

Brackets first: $3 - 7 = -4$. Then $-4 \times (-4) = 16$ (negative times negative is positive).

Divide first: $\dfrac{12}{-3} = -4$. Then $-4 + 5 = 1$.

$(-2)^3 = (-2)\times(-2)\times(-2)$. The first two give $+4$, then $4 \times (-2) = -8$. (An odd power of a negative stays negative.)

The other integer is $-3 - 5 = -8$. (Check: $5 + (-8) = -3$.)

The midpoint is the average: $\dfrac{-6 + 4}{2} = \dfrac{-2}{2} = -1$. (Check: $-1$ is 5 steps from both $-6$ and $4$.)