# Absolute Value The absolute value indicates a [[Complex Numbers|complex number’s]] distance from the origin of the complex plane. ## The Triangle Inequality The triangle inequality states that $ |a+b| \le |a| + |b| $ The only scenario under which $|a+b| = |a| + |b|$ is if the two numbers and the origin of the complex plane all lie on the same line. In all other cases, following the addition rule of complex numbers, the distance from the origin of the sum is less than the sum of original distances. ## $-b \lt a \lt b$ Implies That $|a| \lt b$ We can distinguish between two cases: either $a \ge 0$ or $a \lt 0$. $ \begin{gather} \text{If } a \ge 0 \text{:} & \text{If } a \lt 0 \text{:} \\ |a| = a & |a| = -a\\ \text{since $a \lt b$:} & \text{since $-b \lt a$:} \\ |a| \lt b & b \gt -a\\ & b \gt |a| \end{gather} $ ## $-b \lt a \lt b$ Implies That $b \ge 0$ Lets assume $b \lt 0$. $a$ is either We can distinguish between two cases: either $a \ge 0$ or $a \lt 0$. However $a$ cannot be positive because a positive number can never be bigger than a negative one and we have $a \lt b$. $a$ can also not be negative because we have $-b \lt a$, and we are assuming $b$ to be negative. If $b$ is indeed negative then $-b$ would be positive, and a negative $a$ cannot be bigger than a positive $-b$.