An integer is a whole number that can be positive, negative, or zero, with no fractional or decimal component. Integers include ..., -3, -2, -1, 0, 1, 2, 3, ... and exclude numbers like 0.5, √2, or π.
Integers are fundamental in arithmetic and appear in counting, coding, and financial calculations. They form a subset of real numbers and are distinct from decimals, fractions, and irrational numbers. A number is an integer only if it has no digits after the decimal point and represents a complete unit.
To determine whether a number is an integer, check its format: if it appears as a whole number without a fraction or decimal, it qualifies. For example, 7, -14, and 0 are integers. Numbers like 3.14, 1/2, or √5 are not.
Integers are closed under addition, subtraction, and multiplication. This means adding, subtracting, or multiplying any two integers results in another integer. For example, 5 + (-3) = 2, and 4 × (-6) = -24 - both results are integers.
Division does not preserve integer status. Dividing 6 by 4 gives 1.5, which is not an integer. This property makes integers useful in discrete contexts like inventory counts or loop iterations in programming.
The set of integers is denoted by ℤ, derived from the German word "Zahlen," meaning numbers. This symbol appears in mathematical notation to specify that variables represent whole numbers.
Integers fall into three categories: positive integers (1, 2, 3, ...), negative integers (-1, -2, -3, ...), and zero. Zero is unique because it is neither positive nor negative.
Consider these examples:
| Number | Is It an Integer? | Reason |
|---|---|---|
| 17 | Yes | No decimal or fraction |
| -8 | Yes | Negative whole number |
| 0 | Yes | Neutral whole number |
| 3.14 | No | Has a decimal portion |
| √2 | No | Irrational number, approximately 1.414 |
| 1/3 | No | Fractional value, repeating decimal |
Some numbers look like integers but aren't. For example, 2.0 is mathematically equal to 2, but in programming or data entry, it may be stored as a float. In strict mathematical terms, 2.0 is still an integer because it represents a whole number without remainder.
Many assume that all whole numbers are integers, but the reverse is true: all integers are whole numbers only if they are non-negative. Whole numbers start at 0 and go upward (0, 1, 2, ...), while integers extend infinitely in both directions.
A frequent error is treating repeating decimals as integers. For instance, 0.999... equals 1, which is an integer. But 0.333... (1/3) is not an integer, even though it repeats. Repeating patterns do not imply whole-number status.
Another confusion arises with scientific notation. Numbers like 5e3 (5 × 10³) equal 5000, which is an integer. But 5e-3 (0.005) is not. Always evaluate the actual numeric value, not the format.
Suppose you're given a list: 12, -7, 4.5, 1/2, 0, √16, 3.0. Which are integers?
Four numbers in the list are integers: 12, -7, 0, √16, and 3.0. The others are not. This demonstrates that context matters - √16 must be evaluated to 4 before classification.
During a project involving data validation, I once encountered a CSV file where dates were stored as floating-point timestamps. One entry read 1640995200.0. Though it had a decimal, the value represented midnight January 1, 2022, and was treated as an integer in downstream processing. This shows that format doesn't override mathematical identity.
Integers differ from rational numbers, which include all fractions and terminating or repeating decimals. While 1/2 is rational, it's not an integer. Similarly, real numbers include all rational and irrational numbers, such as π and e, none of which are integers.
For clarity, consider this breakdown:
| Type | Includes Integers? | Examples |
|---|---|---|
| Integers (ℤ) | Yes, by definition | -5, 0, 10 |
| Rational Numbers (ℚ) | Yes, but not all | -3/4, 2.5, 0.333... |
| Real Numbers (ℝ) | Yes, but not all | π, √2, 1.732 |
Every integer is rational and real, but the reverse is false. A number must have no fractional part to qualify as an integer.
According to the National Council of Teachers of Mathematics (NCTM), students often misclassify numbers like 0.999... as non-integers due to their appearance, despite being mathematically equal to 1. This highlights the need for conceptual understanding over visual inspection.
The Common Core State Standards emphasize identifying integers in real-world contexts, such as temperature changes or bank account balances, where only whole numbers represent valid states.
When evaluating a number, ask:
If all answers are "yes" or "no" to the presence of fractional parts, the number is an integer.
Understanding which of the following numbers is an integer requires attention to form and value. Always reduce expressions and examine the result. A number is an integer only if it represents a complete unit with no remainder.