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Understanding Integers and Integer Arithmetic

An integer is any whole number that can be positive, negative, or zero. In mathematical notation, the set of all integers is denoted by the letter $\mathbb{Z}$ (from the German word Zahlen, meaning numbers):

$$\mathbb{Z} = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \}$$

Integers do not include fractions or decimals. Working with integers is one of the most fundamental skills in algebra and pre-calculus, governing how signs interact during addition, subtraction, multiplication, division, and exponentiation.

Rules for Integer Operations

Mastering integer arithmetic requires understanding how positive and negative signs combine across different mathematical operations.

1. Addition of Integers

• Same signs: Add the absolute values of the two numbers and keep the common sign. For example, $(+5) + (+7) = +12$ and $(-4) + (-6) = -10$.
• Different signs: Subtract the smaller absolute value from the larger absolute value, and attach the sign of the integer with the greater absolute value. For instance, $(-12) + 5 = -7$ and $15 + (-6) = 9$.

2. Subtraction of Integers

Subtracting an integer is mathematically identical to adding its additive inverse (its opposite):

$$a - b = a + (-b)$$

For example, $7 - (-4) = 7 + 4 = 11$, and $(-8) - 5 = (-8) + (-5) = -13$. If you need focused practice on step-by-step additions and subtractions, try our Adding and Subtracting Integers Calculator.

3. Multiplication and Division of Integers

When multiplying or dividing two integers:

• Like signs result in a positive answer: $(+) \times (+) = (+)$, $(-) \times (-) = (+)$, $(+) \div (+) = (+)$, $(-) \div (-) = (+)$.
• Unlike signs result in a negative answer: $(+) \times (-) = (-)$, $(-) \times (+) = (-)$, $(+) \div (-) = (-)$, $(-) \div (+) = (-)$.
• Multiplication by zero: Any integer multiplied by zero equals zero ($a \times 0 = 0$).
• Division by zero: Division by zero is undefined in arithmetic ($a \div 0 = \text{undefined}$).

4. Integer Powers and Exponents

When raising a negative integer to an integer power:

• If the exponent is even, the result is positive: $(-3)^2 = 9$, $(-2)^4 = 16$.
• If the exponent is odd, the result is negative: $(-3)^3 = -27$, $(-2)^5 = -32$.

Properties of Integers

Integers possess several foundational mathematical properties:

• Closure: The set of integers is closed under addition, subtraction, and multiplication (the sum, difference, and product of any two integers is always an integer).
• Additive Identity: Zero ($0$) is the additive identity because $a + 0 = a$.
• Additive Inverse: For every integer $a$, there exists an opposite $-a$ such that $a + (-a) = 0$.
• Multiplicative Identity: One ($1$) is the multiplicative identity because $a \times 1 = a$.
• Absolute Value: The absolute value $|a|$ represents the non-negative distance of the integer from zero on the number line.

For integer division with remainders, use our Modulo Calculator, or explore patterns with the Consecutive Integers Calculator.