The set Z

The surface of Lake Tanganyika is well above sea level, but its bottom goes deeper than that level. To write down a position below sea level, a loss or a step backwards, natural numbers are no longer enough: we use integers, positive or negative. In this lesson you learn to compare them and to compute with them.

The set $\mathbb{Z}$

It contains the natural numbers and their opposites: $\mathbb{N}\subset\mathbb{Z}$. The opposite of $5$ is $-5$; the opposite of $-5$ is $5$; $0$ is its own opposite.

The absolute value $|a|$ is the distance from $a$ to $0$: $|-7|=7$ and $|7|=7$.

Comparing: on the number line, the greater number is the one further to the right.

• Every positive number is greater than every negative number.
• Between two negatives, the greater one has the smaller absolute value: $-2>-9$.

Addition and subtraction

• Same sign: add the absolute values and keep the common sign: $(-4)+(-6)=-10$.
• Opposite signs: subtract the smaller absolute value from the larger one and take the sign of the number with the larger absolute value: $(-9)+4=-5$.
• Subtracting means adding the opposite: $a-b=a+(-b)$. So $3-(-8)=3+8=11$.

Bracket rule

• Preceded by $+$: remove them without changing the signs: $+(a-b)=a-b$.
• Preceded by $-$: remove them and change every sign inside: $-(a-b+c)=-a+b-c$.

Multiplication, division and powers

Sign rule (for products and quotients):

Examples: $(-3)\times(-5)=15$; $(-24)\div6=-4$. In $\mathbb{Z}$, the quotient $a\div b$ exists only if the division is exact.

A product of several non-zero factors is negative if the number of negative factors is odd, positive if it is even.

Powers: $(-2)^3=-8$ (odd exponent) and $(-2)^4=16$ (even exponent). Watch out: $-3^2=-(3^2)=-9$, whereas $(-3)^2=9$.

Order of operations: brackets, then powers, then $\times$ and $\div$, then $+$ and $-$.