Rational and Irrational Numbers
Aligned to N-RN.B.3 — Common Core State Standards for Mathematics.
What this lesson teaches
The sum or product of two rational numbers is always rational, since the result can be written as a fraction of integers. But a rational plus an irrational, or a nonzero rational times an irrational, is always irrational.
Worked examples
Example 1
Is 3 + √2 rational or irrational? If 3 + √2 were rational, subtracting the rational 3 would leave √2 as rational too. But √2 is irrational, so 3 + √2 must be irrational.
Example 2
√2 × √2 = 2 — two irrationals can multiply to a rational, so the set is not closed under multiplication.
Example 3
A rational plus a rational is always rational, and a rational times a nonzero rational is too. Only the irrationals misbehave.
Practice questions
- Is 1/4 + 2/3 rational or irrational? Explain.
- Is 5 × √7 rational or irrational? Explain.
- Give an example of two irrational numbers whose sum is rational.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Números Racionales e Irracionales
La suma o el producto de dos números racionales siempre es racional, porque el resultado se puede escribir como una fracción de enteros. Pero un racional más un irracional, o un racional distinto de cero por un irracional, siempre es irracional.
Ejemplo 1: ¿3 + √2 es racional o irracional? Si 3 + √2 fuera racional, al restar el racional 3 quedaría √2 también racional. Pero √2 es irracional, así que 3 + √2 debe ser irracional.
Ejemplo 2: √2 × √2 = 2 — dos irracionales pueden multiplicarse y dar un racional, así que el conjunto no es cerrado bajo la multiplicación.
Ejemplo 3: Un racional más un racional siempre es racional, y un racional por un racional distinto de cero también. Solo los irracionales se portan mal.
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