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Ordering
Integers

Learn to compare and order integers on a number line: from negative to positive, least to greatest, and everything in between.

SSAT Middle Level: Quantitative

What Are Integers?

Integers are the set of whole numbers and their negatives. They include positive numbers, negative numbers, and zero: no fractions, no decimals.

Negative Integers
\( -1, -2, -3, \ldots \)
Less than zero
Zero
\( 0 \)
Neither positive nor negative
Positive Integers
\( 1, 2, 3, \ldots \)
Greater than zero

The Number Line

A number line is a visual tool that places integers in order. Numbers increase as you move right and decrease as you move left.

−5 −4 −3 −2 −1 0 1 2 3 ← Smaller Larger →

Why Ordering Matters

Ordering integers is a foundational skill that appears throughout math: from solving inequalities to interpreting data. On the SSAT, you will see questions that ask you to rank or compare values.

Temperature
Is \( -8^\circ\text{F} \) colder than \( -3^\circ\text{F} \)? Yes: \( -8 \lt -3 \).
Elevation
Death Valley at \( -86\,\text{m} \) is lower than sea level (\( 0\,\text{m} \)).
Money
A debt of \( -\$15 \) means you have less than \( \$0 \).

The Core Rule for Comparing

−3 0 4 −3 < 4

Since \( -3 \) is to the left of \( 4 \), we know \( -3 \lt 4 \). This works for any pair of integers.

Comparing Negative Integers

This is where most students get tripped up. With negative numbers, the one with the larger absolute value is actually smaller.

−7 −2 0 −7 < −2

Magnitude vs. Value

Students often confuse how "big" a number looks with its actual position on the number line. Here is the critical distinction:

Absolute Value (Magnitude) Integer Value (Position)
\( -10 \) \( | {-10} | = 10 \) (large) Very small (far left)
\( -1 \) \( | {-1} | = 1 \) (small) Closer to zero (less small)
\( 5 \) \( |5| = 5 \) (medium) Positive (right of zero)

Comparison Symbols

\( \lt \)
Less than
\( -5 \lt 2 \)
\( \gt \)
Greater than
\( 3 \gt -4 \)
\( = \)
Equal to
\( -6 = -6 \)
\( \leq \) / \( \geq \)
Less/greater than or equal
Includes equality

How to Order a Set of Integers

1
Separate negatives and positives. Group the numbers by sign.
2
Order the negatives: the one with the largest absolute value is the smallest.
3
Place zero in the middle (if present).
4
Order the positives: standard ascending order.
5
Combine: negatives (most negative first), then zero, then positives.

Example 1: Basic Ordering

Order from least to greatest: \( 3,\; -1,\; 0,\; -4,\; 2 \)

−5 −4 −3 −2 −1 0 1 2 3 1st 2nd 3rd 4th 5th

Example 2: All Negatives

Order from least to greatest: \( -2,\; -9,\; -5,\; -1 \)

−10 −9 1st −5 2nd −2 3rd −1 4th

Example 3: Greatest to Least

Order from greatest to least: \( -3,\; 7,\; 0,\; -8,\; 4 \)

1
Identify the largest positive: \( 7 \) is the greatest.
2
Next positive: \( 4 \).
3
Zero comes next.
4
Negatives in descending order: \( -3 \) then \( -8 \).

Example 4: Larger Set

Order from least to greatest: \( 12,\; -15,\; -6,\; 0,\; 8,\; -6,\; 3 \)

1
Negatives: \( -15 \) and \( -6,\; -6 \). Order: \( -15,\; -6,\; -6 \).
2
Zero: \( 0 \).
3
Positives: \( 3,\; 8,\; 12 \).

Writing Inequality Chains

An ordered list can also be written as a chain of inequalities using \( \lt \) or \( \gt \) symbols:

Every symbol in the chain must point the same direction. Mixing \( \lt \) and \( \gt \) in one chain is always wrong.

Common Mistakes

✕ Ignoring the Negative Sign
Thinking \( -7 \gt -2 \) because \( 7 \gt 2 \). Always check position on the number line.
✕ Confusing Direction
Ordering greatest-to-least when the question asks for least-to-greatest, or vice versa. Read carefully.
✕ Forgetting Zero
Skipping \( 0 \) when it appears in the set. Zero is an integer and must be placed between negatives and positives.

Quick Reference Summary

Concept Rule Example
Positive vs. Negative Any positive > any negative \( 1 \gt -100 \)
Two Positives Larger number is greater \( 9 \gt 5 \)
Two Negatives Closer to zero is greater \( -2 \gt -7 \)
Zero Greater than all negatives \( 0 \gt -3 \)
Number Line Right = greater, Left = less Always works

Question 1 of 5

Which list shows the integers ordered from least to greatest?

Question 2 of 5

Which statement is true?

Question 3 of 5

Order from greatest to least: \( -1,\; 4,\; -7,\; 0,\; 3 \)

Question 4 of 5

An integer is greater than \( -6 \) and less than \( -2 \). Which could it be?

Question 5 of 5

The temperatures on five winter mornings were \( -12^\circ, 3^\circ, -5^\circ, -5^\circ, 8^\circ \). Which day was the coldest?