The number no fraction can catch

The idea

A number s is irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.

Fractions, p/q

A fraction is p/q with whole numbers p and q, and q ≠ 0. For example 3/4, or 5 = 5/1.

Can the number √2 be written as p/q?

• Yes, with big enough p and q
• No, never
• I am not sure

The p/q machine

Tap a card. The machine hunts for p and q. Can it find them? Then sort the card.

Irrational numbers

A number s is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.

Examples you know: √2, √3, √15, π, −√2/√3 and 0.10110111011110…

Be careful: √16 = 4 = 4/1 and 22/7 are rational. And a decimal that goes on forever can still be rational: 0.333… = 1/3.

Notes

A number is irrational if it cannot be written as p/q, where p and q are integers and q ≠ 0. Examples: √2, √3, √15, π.

Check yourself

Which of these is irrational?

0.777… goes on forever. Is it irrational?

A friend says: “π = 22/7”. What is true?

Write −2.5 as p/q in the simplest form, with q positive. What is q?

Answer: 2

−2.5 = −25/10 = −5/2. So p = −5 and q = 2.

• √25. √25 = 5 = 5/1. It can be written as p/q.
• 0.5. 0.5 = 1/2.
• √15 — correct. Yes! √15 is one of the irrational examples. No p/q gives exactly √15.
• −7. −7 = −7/1.
• Yes, because it never ends. Never ending is not enough. 7 ÷ 9 = 0.777…, so it is the fraction 7/9.
• No, it equals 7/9 — correct. Yes! 7 ÷ 9 = 0.777… so p = 7 and q = 9. It is rational.
• Cannot tell. We can tell: divide 7 by 9 and see.
• π equals 22/7 exactly. 22/7 = 3.142857… but π = 3.14159265… They are close, not equal.
• π is irrational, and 22/7 is just a nearby fraction — correct. Yes! 22/7 is a fraction, so it is rational. π is irrational: no p/q equals it.
• 22/7 is irrational. 22/7 is already written as p/q with p = 22 and q = 7.