Real Numbers
Main Concept

The real numbers are numbers representing quantities that can vary on a continuous scale. They include the natural numbers, whole numbers, integers, rational numbers and irrational numbers. The set of real numbers is denoted by ℝ.
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Natural numbers: the counting numbers, beginning with 1, represented by ℕ.

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Whole numbers: the natural numbers and 0

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Integers: the natural numbers, their negative counterparts, and 0, represented by ℤ.

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Rational numbers: numbers that can be represented as a ratio of integers; $\frac{\mathrm{a}}{\mathrm{b}}$, where $\mathrm{a}$ and $\mathrm{b}$ are integers and $\mathrm{b}\ne 0$. The decimal representation of a rational number either terminates or has a repeating pattern. The set of rationals is represented by ℚ.

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Irrational numbers: numbers that cannot be represented as a ratio of integers. The decimal form of a rational number is nonrepeating and nonterminating.

Some of the above sets of numbers are subsets, or more specific groups, of other sets. For example, the natural numbers are a subset of the integers; that is, all natural numbers are integers, specifically those integers greater than 0. All integers, in turn, are rational numbers. Given an integer $a$, it can be represented as the ratio $\frac{a}{1}$. On the other hand, the sets of rational and irrational numbers are complements: a number can be either rational or irrational, but not both.



Use the check boxes to see which of the real numbers below are of the given type.
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