You can already add, subtract, multiply, and divide whole numbers using written calculation (columns, carry, and borrow). On a number line, those values lie on or to the right of zero. Numbers can also lie to the left of zero — we call them negative. Temperatures below freezing, bank overdrafts, and positions below sea level are everyday examples of directed numbers.
On a horizontal number line, moving right means the value increases; moving left means it decreases. The distance from zero is the absolute value \(|a|\). For example, \(|-5| = 5\) and \(|3| = 3\).
Comparing
\(a > b\) means \(a\) lies to the right of \(b\). Any negative number is less than any positive number.
Think in terms of movement on the line:
Subtracting a negative is adding a positive: \(5 - (-3) = 5 + 3 = 8\).
| Signs | Result |
|---|---|
| positive × positive, negative × negative | positive |
| positive × negative (either order) | negative |
A ratio compares quantities of the same kind. The ratio \(2:3\) means “2 parts to 3 parts” — five equal parts in total. To share \(40\) in the ratio \(2:3\), one part is \(40 \div 5 = 8\), so the shares are \(16\) and \(24\).
Sharing in ratio \(m:n\)
\[ \text{one part} = \frac{\text{total}}{m+n}, \quad \text{first share} = \frac{m}{m+n}\times\text{total}, \quad \text{second share} = \frac{n}{m+n}\times\text{total} \]
A rate compares different units. Speed is distance per time: \[ \text{speed} = \frac{\text{distance}}{\text{time}}. \] A unit rate tells “how much per one” — for example \(60\) km in \(1.5\) h gives \(60 \div 1.5 = 40\) km/h.
Use the number-line explorer to pick two directed numbers and see their sum on the line. Then try the ratio sharer in the applications section below.