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Rational & irrational numbers

Square roots

Japanese school year: Junior high 3

What you learn

Learn how to classify real numbers by understanding that rational numbers can be expressed as ratios of integers (with non-zero denominators), while irrational numbers cannot. This distinction helps clarify number systems and square root properties. Prior familiarity with fractions and decimals ensures smooth learning.

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Key points

This formula shows the definition of a rational number. Any number that can be expressed as a fraction of integers ab\frac{a}{b} with non-zero denominator bb is rational.

ab(a,b∈Z,  b≠0)\frac{a}{b} \quad (a, b \in \mathbb{Z}, \; b \neq 0)

Use this to convert a repeating decimal with a single repeating digit aa into a fraction. Expressing it as a9\frac{a}{9} shows that repeating decimals are rational numbers.

0.a˙=a90.\dot{a} = \frac{a}{9}

Use this to convert a repeating decimal with a two-digit period into a fraction. Placing the repeating value 10a+b10a + b over 99 expresses it as a rational fraction.

0.a˙b˙=10a+b990.\dot{a}\dot{b} = \frac{10a + b}{99}

Non-terminating, non-repeating decimals (such as 2\sqrt{2} and π\pi) cannot be written as fractions of integers. These numbers are called irrational numbers.

Choose a set to practice.