Absolute Value

Core Idea

The absolute value of a number is its distance from zero on the number line, regardless of direction. It answers the question "how far?" without caring about "which way?" For example, both -5 and 5 are five units from zero, so |−5| = |5| = 5. Absolute value is always non-negative. This concept is important because distance and magnitude come up constantly in mathematics and science — you cannot have a negative distance. It also lays the groundwork for absolute value equations and inequalities in algebra.

How It's Best Learned

Use the number line heavily. Have students count spaces from zero for various integers. Emphasize the language of "distance from zero" rather than "just drop the negative sign," because the latter rule fails when students encounter expressions like |3 − 8| and need to evaluate inside first. Practice with both single numbers and simple expressions inside the bars.

Common Misconceptions

• Students treat absolute value as "make it positive," which leads to errors when the input is an expression (e.g., |x| when x is already positive, or |a − b|).
• Some students think |−3| = −3 because they see the negative sign and don't process the bars.
• Confusing absolute value with "opposite" — the absolute value of a positive number is not its opposite.

Explainer

You have already worked with the number line and learned about opposites: −5 is the opposite of 5, and they sit on opposite sides of zero at equal distances. Absolute value captures exactly that distance. The absolute value of a number is simply how far it is from zero on the number line, measured as a non-negative quantity. Because distance is never negative, absolute value is never negative either. Both 5 and −5 are five steps from zero, so |5| = 5 and |−5| = 5.

Think of it this way: the number line has two directions, positive and negative, but distance does not care about direction — it only cares about magnitude. If you ask "how far is −7 from zero?", the answer is 7, not −7. The bars | | are asking that same question. Everything inside the bars gets evaluated first (like parentheses), and then you report the distance. So |3 − 8| means first compute 3 − 8 = −5, then take the distance: |−5| = 5. This is why "just drop the negative sign" is a dangerous shortcut — it works on bare numbers like |−5|, but fails on expressions.

The formal definition makes this precise: |x| = x when x ≥ 0, and |x| = −x when x < 0. At first, writing |x| = −x looks strange — it seems to say absolute value is negative. But when x is negative, −x is positive (the negative of a negative is positive), so the formula gives the right answer. For example, if x = −5, then |x| = −(−5) = 5. This piecewise definition is exactly what you will use when you solve absolute value equations later, because you will split into two cases based on whether the inside is positive or negative.

Absolute value is foundational because distance is one of the most universal concepts in mathematics. The distance between any two numbers a and b on the number line is |a − b| — regardless of which is larger. Distance between −3 and 7 is |−3 − 7| = |−10| = 10, or equivalently |7 − (−3)| = |10| = 10. Order does not matter. This single idea — absolute value as distance — will reappear in geometry (distance formulas), analysis (how close two quantities are), and statistics (measuring error). Every time you measure how far apart two things are, you are using the concept you learned here.