Integer Operations: Definition, Method and Examples

What are integer operations?

Integer operations are mathematical procedures—specifically addition, subtraction, multiplication, and division—applied to whole numbers, their negative counterparts, and zero.

To perform operations with integers, you must apply specific integer operation rules based on the signs of the numbers involved. Because integers include numbers below zero, these calculations often involve combining values that move in opposite directions on a scale or line.

Integers and their signs

Every integer, except zero, carries a sign that indicates its position relative to zero.

The set of integers includes positive and negative numbers, as well as zero. Positive integers are greater than zero and are typically written without a sign, though they can be written with a plus sign, such as $+5$. Negative integers are less than zero and must always be written with a minus sign, such as $-5$. Zero is strictly neutral, meaning it is neither positive nor negative.

The four operations

Mastering integer arithmetic requires applying different methods to different operations.

When adding integers with the same sign, simply add their absolute values and keep the original sign. When adding integers with different signs, find the difference between their absolute values and keep the sign of the integer that is further from zero. For subtraction, change the subtraction symbol to an addition symbol and immediately flip the sign of the second number.

When multiplying and dividing integers, the rules are even simpler. If both numbers have the exact same sign, the result is always positive. If the numbers have different signs, the result is always negative.

What stays an integer?

Addition, subtraction, and multiplication of integers always produce another integer, but division does not always guarantee an integer result.

This mathematical condition is known as the closure property. If you add, subtract, or multiply any two integers together, the exact result will undoubtedly be an integer. However, dividing one integer by another often leaves a remainder, producing a fraction or decimal. For example, $10÷2=5$ successfully produces an integer, but $10÷4=2.5$ does not.

Properties that help

Mathematical properties of integers allow you to rearrange and simplify expressions before you finish your calculations.

• Commutative property: The order of numbers does not change the result for addition ($a+b=b+a$) or multiplication ($a\times b=b\times a$).
• Associative property: Grouping does not affect the outcome for addition or multiplication. For example, $(a+b)+c=a+(b+c)$.
• Distributive property: Multiplication distributes over addition, meaning $a\times (b+c)=(a\times b)+(a\times c)$.
• Identity property: Adding $0$ leaves an integer unchanged. Multiplying an integer by $1$ also leaves it unchanged.
• Additive inverse property: Adding an integer to its exact opposite always results in $0$, such as $8+(-8)=0$.

Worked examples

These examples show how to apply the rules of integer arithmetic to different operations.

Example 1: Subtracting a negative integer

Question: Evaluate $(-15)-(-6)$.

Method:

1. Keep the first integer exactly as it is.
2. Change the subtraction operation to addition.
3. Change the sign of the second integer to its opposite.
4. Add the newly formed integers: $(-15)+6$.
5. Since the signs are different, subtract the values ($15-6=9$) and keep the sign of the number furthest from zero ($-15$).

Answer: $-9$.

Check: Add the answer to the subtracted amount: $(-9)+(-6)=-15$. This perfectly matches the original starting value.

Example 2: Multiplying integers with different signs

Question: Evaluate $8\times (-4)$.

Method:

1. Multiply the numerical values together as if they were positive: $8\times 4=32$.
2. Check the signs of the original integers.
3. Because one is positive and one is negative, the final product must be negative.

Answer: $-32$.

Check: Divide the final product by the first number: $(-32)÷8=-4$. This perfectly matches the second number.

Example 3: Order of operations with integers

Question: Evaluate $(-20)÷((-2)+6)$.

Method:

1. Solve the expression inside the parentheses first: $(-2)+6$.
2. The signs are different, so find the difference ($6-2=4$) and keep the positive sign.
3. Rewrite the expression with this new result: $(-20)÷4$.
4. Divide the absolute values: $20÷4=5$.
5. A negative divided by a positive is a negative.

Answer: $-5$.

Check: Multiply the final answer by the divisor: $(-5)\times 4=-20$. This matches the original dividend.

Common mistakes

Applying multiplication rules to addition problems is a frequent error when working with integers.

Many learners remember the rule that "two negatives make a positive" and incorrectly apply it to addition. For example, they might write $(-4)+(-3)=7$. However, adding two negative integers means combining two negative movements or deficits, so the result is always negative: $(-4)+(-3)=-7$.

Another common mistake involves dividing by zero. While $0÷5=0$, the expression $5÷0$ is completely undefined and does not represent a valid integer operation.

Frequently asked questions

Is zero an integer?

Yes, zero is an integer. It serves as the exact midpoint between positive and negative numbers on the number line, though it does not carry a positive or negative sign itself.

Are fractions and decimals integers?

No, fractions and decimals are not integers. Integers only include whole numbers, their opposites, and zero. For instance, $4$ and $-7$ are integers, but $4.5$ and $-\frac{1}{2}$ are not.

What are consecutive integers?

Consecutive integers are integers that continuously follow each other in numerical order, increasing by exactly $1$ each time. For example, $-2$, $-1$, $0$, and $1$ form a set of consecutive integers.