Imagine counting from one to ten on your fingers. Each finger represents a distinct number, and you can only have whole fingers—no partial ones. That’s the core idea behind integers in mathematics: they are whole numbers, no fractions allowed.
Integers also include negative numbers. Picture holding your fingers upside down and counting from –1 to –10. Each finger still represents a whole number, and just like you never have a fraction of a finger, you never have a fractional integer. Any number that contains a fraction—whether it’s a simple fraction or a decimal—is not an integer.
Arithmetic—the most basic branch of math—covers addition, subtraction, multiplication, and division. These operations work the same for both positive and negative integers (often called signed numbers). You can also perform arithmetic on absolute values, which means treating all integers as positive regardless of their sign.
Adding Integers – When you add two integers with the same sign, the result keeps that sign and increases in magnitude. If the integers have opposite signs, subtract the smaller absolute value from the larger and keep the sign of the larger number.
Subtracting Integers – Subtracting two integers with the same sign produces a smaller integer. Subtracting a negative integer is equivalent to adding its positive counterpart.
Multiplying and Dividing Integers – If both numbers share the same sign, the result is positive. If their signs differ, the result is negative.
Note that addition and subtraction are inverse operations, as are multiplication and division. For example, adding an integer to zero and then subtracting the same integer returns you to zero. Likewise, multiplying a number by an integer and then dividing by that integer brings you back to the original number.
Every integer can be expressed as a product of prime numbers—integers that cannot be factored further. For instance, 81 equals 3 × 3 × 3 × 3. The Fundamental Theorem of Arithmetic guarantees that this prime decomposition is unique for each integer.
In algebra, letters (variables) stand in for numbers. When a problem specifies that variables represent integers, those variables must be whole numbers. This restriction means you cannot use fractions as values for the variables, although the outcome of operations might still be fractional.
