TMUA Master

You may have first learned modulus as a rule: if x is positive, |x| = x, and if x is negative, |x| = -x.

This is correct, but there is a more useful way to think about modulus: modulus represents distance. In particular, |x| is the distance between x and 0 on the number line.

This immediately explains why modulus can never be negative: distance cannot be negative.

Moving the Reference Point

The reference point does not have to be 0. The expression |x - 3| measures the distance between x and 3.

Consider |x - 3| = 4.

Therefore, |x - 3| = 4 has the two solutions x = -1 and x = 7. More generally, |x - a| = r means that x is exactly r units from a, so when r > 0, x = a - r or x = a + r.

Equal Distances

The distance interpretation becomes especially useful when modulus appears on both sides. Consider |x - 2| = |x - 8|.

Instead of splitting into cases, ask: Which point is equally distant from 2 and 8?

The solution is therefore x = 5.

More generally, |x - a| = |x - b| asks for the point equally distant from a and b. That point is their midpoint, so $x=\frac{a+b}{2}$.

This is not a separate modulus trick. It follows directly from the meaning of modulus as distance.

Modulus Inequalities as Distance

The same viewpoint explains modulus inequalities. Consider |x - 3| < 4 and |x - 3| > 4.

So, for r > 0:

• |x - a| < r means a - r < x < a + r
• |x - a| ≤ r means a - r ≤ x ≤ a + r
• |x - a| > r means x < a - r or x > a + r
• |x - a| ≥ r means x ≤ a - r or x ≥ a + r

The important distinction is simple: less than means inside the distance boundary; greater than means outside it.

From One Dimension to Two Dimensions

Everything so far has taken place on a number line. For real numbers, |x - a| measures distance in one dimension.

In Further Mathematics, the same idea extends to complex numbers. If z = x + iy, then $|z|=\sqrt{x^2+y^2}$, which is the distance from the point (x, y) to the origin in the complex plane.

Modulus therefore develops naturally from distance on a number line to distance in a plane. The meaning stays the same; only the dimension changes.

Key Insight

Think of modulus as distance, not simply as a rule for changing signs:

• |x - a| = r: exactly r units from a
• |x - a| < r: within r units of a
• |x - a| > r: more than r units from a
• |x - a| = |x - b|: equally distant from a and b

This viewpoint turns many modulus equations and inequalities into geometric questions about distance, symmetry, intervals and midpoints, and it prepares the same idea for extension to the complex plane in Further Mathematics.