Numbers on a line can lie to the right of zero (positive) or to the left (negative). Temperatures below freezing, bank overdrafts, and positions below sea level are everyday examples of directed numbers.

The number line

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.

Adding and subtracting

Think in terms of movement on the line:

\(a + b\) — start at \(a\), move \(b\) units (right if \(b > 0\), left if \(b < 0\)).
\(a - b\) — start at \(a\), move \(-b\) units (equivalently, add the opposite of \(b\)).

Subtracting a negative is adding a positive: \(5 - (-3) = 5 + 3 = 8\).

Multiplying and dividing

Signs	Result
positive × positive, negative × negative	positive
positive × negative (either order)	negative

Ratios

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 \(m:n\)

\[ \text{one part} = \frac{\text{total}}{m+n}, \quad \text{first share} = \frac{m}{m+n}\times\text{total}, \quad \text{second share} = \frac{n}{m+n}\times\text{total} \]

Rates

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.

Try it yourself

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.