L1 · Irrational Numbers; the Density of the Number Line

Learning goal

“I can tell a rational number from an irrational number by looking at its decimal, I can write an irrational number as an exact value or as an approximate value, and I can find rational and irrational numbers hiding between any two numbers on the number line.”

Irrational number — a real number that is NOT a quotient of two integers. Its decimal never terminates and never repeats: π = 3.14159265..., √2 = 1.41421356...

Exact value — the number itself, named with a symbol that loses nothing: √2 and π are exact values.

Approximate value — a rounded decimal standing in for the number: √2 ≈ 1.41. The ≈ sign says "approximately equal to."

Density — the number line has no gaps and no "next number": between any two numbers there is always another number, and infinitely many of them.

Worked example

Is 0.12122122212222... rational or irrational? The pattern of 2s grows by one each time: 12, then 122, then 1222... Step 1: A rational number's decimal does one of two things - it terminates (0.375) or it repeats one fixed block forever (0.121212... repeats "12"). Step 2: This decimal never terminates, and no fixed block repeats: the block of 2s keeps growing, so the decimal can never settle into a repeating pattern. Step 3: A decimal that never terminates and never repeats is irrational - it is not the decimal of any fraction a/b. Say it plainly: classify by the DECIMAL'S BEHAVIOUR, not by whether the number "looks strange." π is the most famous irrational: π = 3.1415926535..., non-terminating and non-repeating. Written exactly it is just π; written as 3.1416 it is an approximation.

Check yourself — explain it

Why is 0.121212... rational but 0.121221222... irrational?

Paired problem — example still visible

Is √9 rational or irrational? -> Rational. √9 = 3 exactly, because 3 x 3 = 9 - and 3 = 3/1 is a quotient of integers. The √ costume does not decide; the value does.

Faded completion — last step is yours

"π = 3.14159265... never terminates and never repeats, so π is an ___________ number, and 3.1416 is its ___________ value." -> irrational; approximate

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