Properties of Absolute Value

Algebraic Properties Beyond Products

Absolute value turns a real number into its nonnegative magnitude. The Basic Properties of Absolute Value theorem established that absolute value is nonnegative, is zero exactly at zero, and satisfies \(|-x|=|x|\). The Absolute Value of a Product theorem established that \(|xy|=|x||y|\). These facts allow us to analyze quotients and powers without repeatedly returning to the piecewise definition.

Division requires one additional condition: the denominator must be nonzero. Once that condition is in place, absolute value of a quotient behaves just as expected. The reason is that division is multiplication by a reciprocal, and the absolute value of a reciprocal is the reciprocal of the absolute value.

Proof. Since \(y\ne0\), its reciprocal \(y^{-1}\) exists, and \(\frac{x}{y}=xy^{-1}\). By the Absolute Value of a Product theorem,

Apply the product theorem to \(yy^{-1}=1\). Since \(|1|=1\), we obtain

Because \(y\ne0\), the Basic Properties of Absolute Value theorem gives \(|y|>0\). Dividing the last equality by \(|y|\) therefore gives \(|y^{-1}|=1/|y|\). Substitution yields

This proves the identity, including the case \(x=0\). \(\square\)

Comparing Magnitudes by Comparing Squares

Absolute value removes the sign of a number, while squaring also removes its sign. This parallel makes squares a convenient way to compare magnitudes. The comparison is valid because absolute values are nonnegative: on nonnegative real numbers, squaring preserves order. We first record the resulting equivalence for arbitrary real numbers.

Proof. By the definition of absolute value, \(|x|^2=x^2\): if \(x\geq0\), then \(|x|=x\); if \(x<0\), then \(|x|=-x\), and \((-x)^2=x^2\). The same argument gives \(|y|^2=y^2\).

Suppose first that \(|x|\leq|y|\). Both sides are nonnegative. The theorem on multiplying nonnegative inequalities, applied with \(0\leq |x|\leq |y|\) in both factors, gives

Using \(|x|^2=x^2\) and \(|y|^2=y^2\), this is \(x^2\leq y^2\).

Conversely, suppose \(x^2\leq y^2\), so \(|x|^2\leq|y|^2\). If \(|x|>|y|\), then \(|x|-|y|>0\) and \(|x|+|y|>0\). Their product is positive, so

That contradicts \(|x|^2\leq|y|^2\). Therefore \(|x|\leq|y|\), completing the proof. \(\square\)

The strict version follows as well: \(|x|<|y|\) if and only if \(x^2<y^2\). One direction follows by multiplying positive quantities, and the reverse follows from the non-strict equivalence together with the fact that equality of the squares would force equality of the absolute values. In practice, the non-strict result is often enough: it replaces a comparison involving absolute values with a comparison of ordinary squares.

The Reverse Triangle Inequality

The Triangle Inequality bounds the magnitude of a sum from above. A complementary estimate bounds how far apart two magnitudes can be. This is called the reverse triangle inequality. It does not say that absolute value preserves differences exactly; rather, it says that changing a number by a small amount cannot change its magnitude by more than that amount.

Proof. Write \(x=(x-y)+y\). By the Triangle Inequality,

Subtracting \(|y|\) from both sides gives \(|x|-|y|\leq|x-y|\). Interchanging \(x\) and \(y\) in the same argument gives

where \(|y-x|=|x-y|\) follows from \(|-u|=|u|\). Together, the two inequalities say that the real number \(|x|-|y|\) lies between \(-|x-y|\) and \(|x-y|\):

Since \(|x-y|\geq0\), the Absolute-Value Bound Criterion gives \(\bigl||x|-|y|\bigr|\leq|x-y|\). This proves the result. \(\square\)

The estimate is useful when the exact values of two magnitudes are inconvenient, but the difference between the underlying numbers is easy to control. It also applies when \(x\) and \(y\) have opposite signs. In that case, their magnitudes may change in a different way than when they have the same sign, but the bound remains valid.

What Absolute Value Does Not Preserve

Absolute value preserves products and quotients (when defined), but it does not preserve ordinary order. For example, \(-4<2\), whereas \(|-4|=4\) is not less than \(|2|=2\). The comparison theorem above gives the correct way to compare magnitudes: compare the squares, or first determine the signs and then compare the nonnegative absolute values. Replacing an inequality by one involving absolute values without checking signs can reverse or destroy the comparison.

There is a related distinction between \(|x-y|\) and \(|x|-|y|\). The first is the distance between \(x\) and \(y\); the second is a signed difference between their magnitudes. The reverse triangle inequality compares them by saying that the absolute value of the second quantity cannot exceed the first. It does not generally assert equality. For example, with \(x=5\) and \(y=-2\), the left side is \(\bigl|5-2\bigr|=3\), while the right side is \(|5-(-2)|=7\).

These properties work together with the results already established: use the quotient identity for division, compare squares when deciding which magnitude is larger, and use the reverse triangle inequality when estimating a change in magnitude. In each case, check the hypotheses first, particularly that a denominator is nonzero and that any order comparison is being made between nonnegative quantities.