Integers

Compute $-8 + 5$.

Compute $-4 - (-9)$.

Compute $(-3) \cdot (-4) \cdot (-1)$.

Compute $(-6) + (-7)$.

In the proof that $(-a)(-b) = ab$, we wrote $(-a)(-b) + (-a)b = (-a)\big((-b) + b\big)$. Which property justifies this step?

For which integers $a$ is there an integer $b$ with $a \cdot b = 1$? (Choose all that apply.)

At dawn in Ulaanbaatar the temperature is $-23\,^\circ\mathrm{C}$. By noon it has risen by $15$ degrees. What is the noon temperature, in $^\circ\mathrm{C}$?

Write find_mult_partner(a, limit) that searches the integers $b$ with $-\text{limit} \le b \le \text{limit}$ and returns the first $b$ with a * b == 1, or None if there is none. Then think: why does searching further never help when $|a| \ge 2$?