The Other Side of Zero
Explore positive and negative numbers, integers, Bela's Building of Fun, elevator movement rules, token models, additive inverses, and arithmetic on the number line.
10.0 More and More Numbers!
Recall that the very first numbers we learned about in mathematics were the counting numbers $1, 2, 3, 4, \dots$. Then we learned about $0$ (zero), representing nothing, followed by fractions like $\frac{1}{2}$ and $\frac{3}{2}$.
Are there numbers that come before $0$ and are less than $0$? We know the standard number ray starts at $0$ and goes forever to the right. Completing this ray to the left gives us negative numbers, forming a complete number line.
10.1 Bela's Building of Fun
Bela purchased a multi-storied building with attractions for children. Entry to the building is at the ground floor level, called the Welcome Hall (Floor 0).
- Floors above the ground floor are numbered with positive numbers ($+1, +2, +3, \dots$).
- Floors below the ground floor are numbered with negative numbers ($-1, -2, -3, \dots$).
- Zero ($0$) is neither positive nor negative.
Interactive Lift Simulator
Select buttons to move between floors in Bela's Building of Fun:
Current Floor: Floor 0 (Welcome Hall)
Expression: $0$
10.2 Additive Inverses & Number Line
When you press $+3$ followed by $-3$, you return to the ground floor ($0$). Therefore, $-3$ is the additive inverse of $+3$, and $+3$ is the inverse of $-3$.
Movement Formula
$$\text{Starting Floor} + \text{Movement} = \text{Target Floor}$$ $$\text{Target Floor} - \text{Starting Floor} = \text{Movement Needed}$$10.3 The Token Model
We can represent integer operations using green positive tokens + and red negative tokens -.
Combine 5 positive tokens and 3 negative tokens:
+++++ + ---
Canceling 3 zero pairs leaves 2 positive tokens: $+2$.
Chapter Summary
- Integers consist of positive numbers, negative numbers, and zero: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.
- Zero ($0$) is neither positive nor negative.
- For any integer $a$, its additive inverse is $-a$ such that $a + (-a) = 0$.
- Numbers to the right on a number line are greater than numbers to the left (e.g., $-3 < 2$ and $-5 < -3$).
- Subtracting an integer is the same as adding its inverse.
Solved Exercises (Figure It Out)
a. $(+1) + (+4) = \mathbf{+5}$
b. $(+4) + (+1) = \mathbf{+5}$
c. $(+4) + (-3) = \mathbf{+1}$
d. $(-1) + (+2) = \mathbf{+1}$
e. $(-1) + (+1) = \mathbf{0}$
f. $0 + (+2) = \mathbf{+2}$
g. $0 + (-2) = \mathbf{-2}$
a. $-2 \;\mathbf{<}\; +5$
b. $-5 \;\mathbf{<}\; +4$
c. $-5 \;\mathbf{<}\; -3$
d. $+6 \;\mathbf{>}\; -6$
e. $0 \;\mathbf{>}\; -4$
f. $0 \;\mathbf{<}\; +4$
a. $(+1) - (+4) = \mathbf{-3}$
b. $(0) - (+2) = \mathbf{-2}$
c. $(+4) - (+1) = \mathbf{+3}$
d. $(0) - (-2) = \mathbf{+2}$
e. $(+4) - (-3) = \mathbf{+7}$
f. $(-4) - (-3) = \mathbf{-1}$