Question

Solve the equation. $$|a|=0$$

Short answer

The solution to the equation \(|a| = 0\) is \(a = 0\).

Step-by-step solution

Understand the Absolute Value

Firstly, remember that the absolute value of a number is its distance from 0 on a number line. It is always non-negative. The absolute value of \(a\) is symbolized as \(|a|\). This can either be positive when \(a\) is positive or zero if \(a\) is zero itself.

Solving the equation \[|a|=0\]

Since the absolute value of a number can only be non-negative (i.e., positive or zero), the only solution for |a| = 0 is when a = 0. This is because any non-zero number would produce a positive absolute value.

Key concepts

Absolute Value

The concept of absolute value is fundamental in understanding various forms of algebraic equations. It refers to the distance of a number from zero on the number line, regardless of the direction. Think of it like the 'numerical value' without any concern for whether it's originally positive or negative. For instance, the absolute value of both -5 and 5 is 5 because each is five units away from zero on the number line.

When dealing with absolute values, remember these essential characteristics: they are always non-negative, because distance cannot be negative; and the absolute value of zero is zero itself, because there is no distance between zero and zero. This explains why in our original equation, \(|a| = 0\), the only viable solution is when \(a\) is exactly zero.

Solving Algebraic Equations

Algebraic equations can sometimes appear complex, but solving them often involves understanding and applying a few key principles simply and systematically. When you see an equation like \(|a| = 0\), it's essentially a puzzle where you're tasked to find the value of \(a\) that makes the statement true. Remember that to solve an equation means to find all values of the variable that satisfy the equation.

In the case of absolute value equations, there's a specific approach. Since the absolute value is the distance from zero, you often end up with two possible solutions when the other side of the equation is not zero: one positive and one negative. But with \(|a| = 0\), there's only one solution because zero is the only non-negative number that has no distance from itself.

Non-negative Numbers

In mathematics, we refer to numbers that are either greater than or equal to zero as non-negative numbers. This set includes all positive numbers and zero itself. They are crucial when discussing absolute values because, by definition, an absolute value cannot be negative. It represents a magnitude or size, and magnitudes are inherently positive or zero.

Whether you are adding, subtracting, multiplying, or dividing, if you are working with non-negative numbers, the result retains these properties. This intrinsic characteristic of non-negative numbers ensures that the absolute value of any number, or the solution to an equation like ours, \(|a| = 0\), will always be a non-negative number. The concept of non-negativity reinforces the idea that if the absolute value of a number is zero, the number itself must indeed be zero.