You can order numbers by their position on the line. Sometimes we care instead about how far a number is from 0, not which side it is on. You saw this with opposites: 3 and $-3$ are the same distance from 0.
Idea clave
The absolute value of a number is its distance from 0, written $|a|$. Distance is never negative: $|-5| = 5$ and $|5| = 5$.
Por qué funciona
Absolute value counts steps from 0, and you cannot walk a negative number of steps. So it keeps the size of a number and forgets its direction. That is why it is useful for things like the size of a debt or how far below sea level a submarine is.
Ejemplo resuelto: Find the absolute value
Find $|-7.2|$ and $|0|$.
- $-7.2$ is 7.2 units from 0.
- 0 is 0 units from itself.
Respuesta: $|-7.2| = 7.2$ and $|0| = 0$.
Ejemplo resuelto: Which numbers?
Find every number $x$ with $|x| = 6$.
- We need every number 6 units from 0.
- One is 6 units right of 0 and one is 6 units left.
Respuesta: $x = 6$ or $x = -6$.
Ejemplo resuelto: Bigger debt, smaller balance
Jo’s balance is $-30$ dollars and Ana’s is $-10$ dollars. Who owes more?
- A negative balance is a debt. The size of the debt is the absolute value: $|-30| = 30$ and $|-10| = 10$.
- Jo owes \$30 and Ana owes \$10.
Respuesta: Jo owes more, even though $-30 < -10$.
| Balance | Compared as numbers | Size of the debt |
|---|---|---|
| $-30$ | $-30 < -10$ | $|-30| = 30$ |
| $-10$ | $|-10| = 10$ |
Error común
Mixing up order and size. "$-30$ is less than $-10$, so it is less debt" sounds right but is backward: $-30$ is less as a number (farther left), yet its size, the debt, is bigger. Quick check: ask whether the question is about position ("which is less?") or size ("how much is owed?"). Size questions need absolute value.