Key takeaway: The real numbers are all the rational and irrational numbers together, which means every point on the number line corresponds to exactly one real number.
Grade 88.NS.A.1Numbers and Operations
Definition of Real Numbers
The real numbers are the complete set of values you can place on a number line. Counting numbers, zero, negatives, fractions, terminating and repeating decimals, square roots such as \(\sqrt{50}\), and π all belong. Between any two real values there is always another one, so the line has no gaps.
The set splits cleanly into two families. Rational numbers can be written as a fraction of integers. Irrational numbers cannot. Nothing is in both, and nothing real is outside both. This structure lets you compare any two real numbers by size, even when one is a fraction and the other is a root, by converting each to a decimal estimate.
Formula and notation
The set of real numbers has its own symbol, and it splits into rational and irrational parts:
\[\mathbb{R} = \mathbb{Q} \cup \{\text{irrational numbers}\}\]
Smaller sets nest inside: natural numbers, whole numbers, integers, then rationals.
Key facts
- Every real number has a decimal expansion, which may end, repeat or neither.
- No number is both rational and irrational.
- Real numbers can be ordered, so for any two of them one is larger or they are equal.
- The square root of a negative number is not real; it needs imaginary numbers.
Real Numbers example
Place these numbers in order from least to greatest and say which are irrational: \(\sqrt{64}\), \(-6\), \(0.\overline{7}\), \(\sqrt{50}\), \(\frac{5}{8}\).
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Convert each to a decimal: \(\sqrt{64} = 8\); \(-6\); \(0.\overline{7} \approx 0.778\); \(\sqrt{50} \approx 7.07\); \(\frac{5}{8} = 0.625\).
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Compare: \(-6 < 0.625 < 0.778 < 7.07 < 8\).
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Only \(\sqrt{50}\) is irrational, since 50 is not a perfect square.
The order is \(-6,\ \frac{5}{8},\ 0.\overline{7},\ \sqrt{50},\ \sqrt{64}\), and the only irrational value is \(\sqrt{50}\).
Common mistake
Watch out
Wade says that \(-4\) is not one of the real numbers, because “real things cannot be negative.”
Why it is wrong
Real in math does not mean a physical object. The set covers everything on the number line, including all negatives and zero. The value \(-4\) sits four units left of zero, so it is a real number, an integer and a rational number at once.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Numbers and Operations