Real Numbers
In short: Real numbers are all the values that sit on the number line, combining the rational numbers (fractions and their decimals) with the irrational numbers such as pi.
8th GradeNumbers and Operations8.NS.A.1
What is Real Numbers?
Every point on a number line corresponds to one of the real numbers, and every real number corresponds to a point. The set covers integers, fractions, terminating and repeating decimals, and irrational values such as \(\sqrt{5}\) and \(\pi\).
Classification nests like boxes: natural numbers sit inside whole numbers, which sit inside integers, which sit inside rational numbers. The irrational numbers stand outside the rational box, and together the two fill the line with no gaps. Any real number is one or the other, never both.
How to spot it
- A problem asks which sets a number belongs to: whole, integer, rational, irrational or real.
- Numbers of mixed kinds, like negative fractions, roots and decimals, must be ordered or compared.
- A statement claims that a number is real, or asks whether a square root of a negative is on the line.
- You are asked to place values such as the root of 5 or pi on a number line.
The method
- Rewrite each number as a decimal or a simple fraction so its type is visible.
- Check from the innermost set outward: natural, whole, integer, rational.
- If no fraction of integers matches it, call it irrational; both rational and irrational are real.
- To order mixed numbers, convert them all to decimals, then place them on a line.
Real Numbers: solved example
Problem
Classify each number by the most specific set: \(-7\), \(\frac{2}{3}\), \(\sqrt{5}\), \(0\), \(4.2\), \(\pi\).
Solution
\(-7\) is an integer; \(\frac{2}{3}\) is rational (repeating decimal \(0.\overline{6}\)); \(0\) is a whole number; \(4.2 = \frac{21}{5}\) is rational.
\(\sqrt{5}\) and \(\pi\) are irrational. All six are real numbers, so every one of them lies on the number line.
Try it
Question 1
Order from least to greatest: \(-\sqrt{3}\), \(0.5\), \(\frac{4}{3}\).
Show the answer
\(-\sqrt{3} \approx -1.732\), \(0.5\), \(\frac{4}{3} \approx 1.333\). The order is \(-\sqrt{3},\ 0.5,\ \frac{4}{3}\).
Question 2
Can a real number be both rational and irrational?
Show the answer
No. A real number either can be written as a fraction of integers or cannot, so the two sets do not overlap.
Real Numbers vs. Irrational Number
The irrational numbers are one part of the real numbers: those that no fraction can represent. The real numbers are the entire family, which also includes every integer, fraction and terminating or repeating decimal. Calling a number real says very little; calling it irrational says something specific.
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Whole dictionary: A–Z · 8th Grade · Numbers and Operations