Comparing and Ordering Integers

One rule decides every comparison

Whatever sits further right on the number line is greater.

That rule is the whole topic. It works for positives, for negatives, for fractions and for any mixture of them.

$$5 > 2 \qquad 2 > -1 \qquad -1 > -4$$

Each of those says the same thing: the first number sits to the right of the second.

Why $-8 < -3$

This is the comparison that trips people up.

$-8$ sits further left, so it is less than $-3$:

$$-8 < -3$$

The digits pull the other way, and that is the trap. $8$ is bigger than $3$, so $-8$ looks like it should be bigger too. But the minus sign sends it in the opposite direction — further from zero on the left, which is further down.

A thermometer settles it. $-8$ degrees is colder than $-3$ degrees.

Reading the inequality signs

Sign	Means	Example
$<$	is less than	$-7 < -2$
$>$	is greater than	$3 > -5$
$\le$	is less than or equal to	$-4 \le -4$
$\ge$	is greater than or equal to	$0 \ge -1$

The wide end faces the larger number and the point faces the smaller one.

An inequality is a statement about position. $-7 < -2$ says exactly one thing: $-7$ sits left of $-2$ on the line.

Two facts that shortcut most questions

Every negative is less than every positive. Negatives live left of zero and positives live right of it, so no comparison between them needs any thought:

$$-1000 < 1$$

Zero sits between them. Any negative is less than zero, and any positive is greater:

$$-3 < 0 < 3$$

Together these split any list into three groups before you compare anything within them.

Ordering a mixed list

Put everything on one line and read left to right. Converting fractions to decimals makes the positions easier to see.

Order $-2, \; \tfrac{1}{2}, \; -3.5, \; 1, \; -0.25$.

Negatives first, then zero-ish values, then positives:

$$-3.5 \; < \; -2 \; < \; -0.25 \; < \; \tfrac{1}{2} \; < \; 1$$

Note that $-0.25$ is the largest of the three negatives, because it is the closest to zero.

Why the direction of the rule matters in context

The number-line rule and everyday language can point opposite ways, and knowing which is which prevents a genuine misreading.

A temperature of $-15$ is less than $-4$. Everyday speech calls $-15$ “more extreme” or “a bigger drop”, and both are fair descriptions of the weather — but as numbers, $-15 < -4$.

The same split shows up with money. A debt of $200$ is a bigger debt than a debt of $50$, and as balances, $-200 < -50$. The larger debt is the smaller number.

“Bigger” in conversation often means further from zero. “Greater” in mathematics always means further right. Once a question is written with $<$ or $>$, only the second meaning applies.

Worked examples

Common mistakes

Practice problems