Absolute Value and Ordering Integers

Absolute value tells you how far a number is from zero, and ordering integers means placing them from least to greatest (or greatest to least) on a number line.

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Definition

An integer is any whole number, its opposite, or zero — like -5, 0, or 8. The absolute value of an integer is its distance from zero on a number line, and distance is always zero or positive, never negative. Ordering integers means lining them up in order by their value, where numbers to the right on a number line are always greater than numbers to the left.

Remember the rule

Absolute value rule: |n| = n if n is positive or zero; |n| = -n if n is negative. On a number line, always read left to right for least to greatest — more negative means smaller value!

Key words

Integer: Any whole number that can be positive, negative, or zero — for example, -10, -3, 0, 4, 17.
Absolute Value: How far a number is from zero on a number line. Written with two vertical bars like |−6|. The answer is always 0 or positive.
Opposite: Two numbers the same distance from zero but on different sides — like -5 and 5. They are opposites of each other.
Number Line: A straight line with numbers placed at equal spaces. Negative numbers go left, positive numbers go right, and zero is in the middle.
Negative Integer: A whole number less than zero, written with a minus sign in front, like -3 or -100.
Positive Integer: A whole number greater than zero, like 2, 15, or 99.
Inequality: A math statement that shows one number is greater than or less than another, using the symbols > (greater than) or < (less than).
Ordering: Arranging a set of numbers from least to greatest or from greatest to least.

Worked examples

What is |−8|?

→ 8 · −8 is 8 steps away from zero on the number line, so its absolute value is 8.

What is |5|?

→ 5 · 5 is already positive — it is 5 steps from zero, so |5| = 5.

What is |0|?

→ 0 · Zero is already at zero, so its distance from zero is 0.

Order these integers from least to greatest: 3, -7, 0, -2, 5

→ -7, -2, 0, 3, 5 · Plot them on a number line — the farthest left is smallest. Negatives always come before zero and positives.

Which is greater: -10 or -3?

→ -3 is greater than -10 · On a number line, -3 is to the right of -10, so -3 > -10 even though 10 looks like a bigger number.

List these from greatest to least: |-4|, |2|, |-9|, |0|

→ 9, 4, 2, 0 · First find each absolute value: |-4|=4, |2|=2, |-9|=9, |0|=0. Then order those results from greatest to least.

Common mistakes

Thinking absolute value can be negative — it is ALWAYS zero or positive, so |-7| = 7, never -7.
Assuming a bigger-looking negative number is greater — -10 is actually LESS than -3 because it is farther left on the number line.
Forgetting that zero is neither positive nor negative, but its absolute value is 0.
Confusing the absolute value bars | | with the number 1 or with parentheses — they mean distance from zero, not multiplication or grouping.
When ordering integers with absolute values, solving the absolute values first before ordering, instead of ordering the original integers — make sure you know which task is being asked.

FAQs

Can absolute value ever be a negative number?

No, never. Absolute value is a distance, and you cannot have a negative distance. The result is always zero or a positive number.

Why is -10 less than -3 if 10 is bigger than 3?

Think of a number line: -10 is farther to the LEFT than -3. The farther left a number is, the smaller it is. Owing 10 dollars is worse than owing 3 dollars!

Is the absolute value of a positive number just the same number?

Yes! |7| = 7. Absolute value only changes things for negative numbers — it flips them positive. Positive numbers and zero stay the same.

How do I order a mix of positive and negative integers quickly?

Put all the negatives on the left (most negative first), then zero in the middle, then all the positives on the right (smallest positive first). A number line sketch always helps!

What does it mean when a problem says 'order by absolute value'?

First find the absolute value of each number, then order those results. For example, ordering -5, 2, -1 by absolute value gives 5, 2, 1 — so the order from greatest absolute value is -5, 2, -1.

Are -4 and 4 the same because they have the same absolute value?

They have the same absolute value (both equal 4), but they are NOT the same number. -4 is less than 4. Same distance from zero, but on opposite sides.

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