Add, subtract, multiply, and divide integers.

For thousands of years, mathematicians struggled with the very concept of a negative number. Ancient Greek mathematicians refused to accept them entirely, viewing mathematics as a study of magnitudes — lengths, areas, and volumes — which could never be less than zero. The idea that you could owe a debt, fall below a reference point, or measure a temperature below freezing eventually forced civilizations to expand their number systems. The story of integer arithmetic is really the story of humanity learning to count in both directions.

Today, integer operations are foundational to algebra, coordinate geometry, and virtually every quantitative discipline. On the ISEE, questions involving integers appear throughout the Mathematics Achievement section, sometimes disguised inside more complex problems. If you can handle sign rules confidently, you remove an entire category of careless errors from your test.

An integer is any whole number — positive, negative, or zero. The set of integers is written { … , −3, −2, −1, 0, 1, 2, 3, … }. Unlike fractions or decimals, integers have no fractional part. Every integer has an absolute value, which is its distance from zero on the number line, always expressed as a non-negative number. For example, |−5| = 5 and |5| = 5. Understanding absolute value is the key to mastering all four operations with integers.

A number line is one of the most powerful tools for understanding integer arithmetic. Every addition can be seen as a movement — positive means move right, negative means move left. Subtraction reverses the direction of the second number. The diagram below illustrates two addition examples: one with same signs and one with different signs.

When the ISEE presents tricky sign combinations, mentally picturing this number-line movement can prevent careless mistakes. Adding a negative is the same as moving left; subtracting a negative reverses the direction, sending you right. If you ever feel confused by a string of signs, pause, visualize the direction, and the answer will become clear.

Every integer operation boils down to two questions: what is the magnitude of the result, and what is the sign? The equations below formalize the rules you saw on the concept grid. Memorizing these patterns will let you work through ISEE problems quickly and confidently.

The chart below consolidates every sign outcome for all four operations in one place. Study it until the patterns become automatic — the ISEE rarely tests sign rules directly, but nearly every algebra question depends on getting them right behind the scenes.

Common Pitfalls on the ISEE

• Double-negative confusion: Students see "−(−5)" and freeze. Remember: subtracting a negative is the same as adding a positive. Write it out: −(−5) = +5.
• Forgetting zero's neutrality: Zero is neither positive nor negative. Adding or subtracting zero changes nothing. Multiplying by zero always gives zero.
• Confusing exponent signs: (−3)² = 9, but −3² = −9. The parentheses make all the difference — the exponent applies only to what's inside them.
• Rushing through chains: In expressions like (−2)(−3)(−4), count the negatives first (three negatives → odd → negative), then compute the magnitude: 2 × 3 × 4 = 24, so the answer is −24.

Let's work through a multi-step problem that combines several integer operations, just as the ISEE might present it. The key is to handle one operation at a time, applying sign rules carefully at each stage.

One reason integer problems feel tricky is that the same pair of numbers can produce very different results depending on the operation. The table below shows how changing the operation changes the outcome for the same two integers. Studying these contrasts trains your brain to slow down and check the operation before jumping to a sign rule.

Integer operations aren't just a standalone topic — they are the engine that drives nearly every algebraic skill tested on the ISEE. When you solve equations, simplify expressions, or evaluate functions, you are performing integer arithmetic at every step. The table below maps how each integer skill connects to higher-level concepts you'll encounter on the test.

As you move into more advanced mathematics — coordinate geometry, systems of equations, and eventually calculus — you'll find that integers and their sign rules remain constant. Mastering them now creates a foundation of speed and accuracy that pays dividends across every future math course.

Work through these five problems in order. Each one builds on the skills covered in this lesson. Remember: on the ISEE, there is no penalty for guessing, so always answer every question. Use process of elimination to rule out wrong choices before picking your answer.