Once numbers are on a number line, ordering them is a matter of looking left and right.
Key idea
Numbers increase from left to right. Farther left is less: $-8 < -3$, even though 8 is greater than 3.
Why it works
Think of temperatures. $-8$ degrees is 8 degrees below zero and $-3$ degrees is 3 below zero, so $-8$ is colder: it is less. For negative numbers, a bigger distance from 0 means a smaller number, because you have gone farther in the "less" direction.
Worked example: Compare two negatives
Which is greater, $-1.5$ or $-1.25$?
- $-1.5$ is 1.5 units left of 0; $-1.25$ is 1.25 units left of 0.
- $-1.25$ is closer to 0, so it is farther right.
Answer: $-1.25 > -1.5$
Worked example: Order a list
Write $-1.5$, 2, $-\frac{3}{4}$ and $-2$ from least to greatest.
- The negatives come first. Of $-2$, $-1.5$ and $-\frac{3}{4}$, the one farthest from 0 is the least: $-2$.
- Next $-1.5$, then $-\frac{3}{4}$ (which is $-0.75$).
- The positive number comes last.
Answer: $-2 < -1.5 < -\frac{3}{4} < 2$
Worked example: In a story
At dawn it was $-12$ °F in Fargo and $-7$ °F in Duluth. Write an inequality and say which city was colder.
- $-12$ is farther below 0 than $-7$.
- So $-12 < -7$.
Answer: $-12 < -7$: Fargo was colder.
Common mistake
Saying $-8 > -3$ because $8 > 3$. That compares the distances from 0, not the numbers. Quick check: picture a thermometer. Which reading is lower? The lower one is the lesser number.