B. Number · Number Sense: Scientific Notation, Real Numbers, Square Roots and Percents

Irrational Numbers: Decimals That Never Repeat

B1.2About 40 minutes

Learning goals

I can tell from its decimal whether a number is rational or irrational.
I can explain why √2 and π are irrational and why 3.14 and 22/7 are only approximations of π.
I can show where √2 is on a number line.
I can tell whether the square root of a whole number is rational or irrational.

Words to know

terminating decimal
A decimal that ends, like 0.625.

repeating decimal
A decimal in which a block of digits repeats forever, like 0.272 727…

non-repeating, non-terminating decimal
A decimal that never ends and never settles into a repeating block. This is the decimal of an irrational number.

approximation
A value that is close to the exact value but not equal to it. 3.14 is an approximation of π.

Remember from Grade 7

In Grade 7 you changed fractions to decimals by dividing. Some decimals terminate (3/8 = 0.375) and some repeat (5/6 = 0.833 3…). You learned to predict which: in simplest form, if the denominator has only 2s and 5s as prime factors, the decimal terminates.

Two kinds of decimals, or three?

Every rational number has a decimal that either terminates or repeats. Why? When you divide a by b, the remainders can only be 0, 1, 2, … up to b − 1. If you get remainder 0, the decimal ends. If not, sooner or later a remainder must come back, and from then on the digits repeat.

So a decimal that goes on forever without repeating can't come from a fraction. It belongs to an irrational number.

Dividing 1 by 7: each remainder (big circle) gives the next digit (small circle). Only 6 non-zero remainders are possible, so they go round in a loop and 1/7 = 0.142857… repeats.

Decimal type	Example	Rational?
terminates	0.625 = 5/8	yes
repeats	0.272 727… = 3/11	yes
never ends, never repeats	1.414 213 562… = √2	no: irrational

Every decimal is one of these three types.

Famous irrational numbers

√2 = 1.414 213 562 3… If you square 1.414 you get 1.999 396, close to 2 but not 2. No decimal you can write, and no fraction, squares to exactly 2.
π = 3.141 592 653 5… π is the circumference of any circle divided by its diameter. People have calculated trillions of digits of π, and no repeating block has ever appeared. Mathematicians have proven that π is irrational.
The square root of any whole number that is not a perfect square is irrational: √3, √5, √10, √50, …

What about 3.14 and 22/7? Both are rational approximations of π. 3.14 = 314/100, and 22/7 = 3.142 857 142 857…, which repeats. They are close to π, but not equal to it.

A square with sides of 1 unit has a diagonal of √2 units. Swing the diagonal down onto the number line (dashed arc) and it lands at √2 ≈ 1.41, a little less than halfway from 1 to 2.

Worked example

Example 1: Terminating, repeating or neither?

Is 5/9 rational? What is its decimal?

5 ÷ 9 = 0.555 5… The 5 repeats forever.
It is a fraction of integers, and its decimal repeats, which matches.

5/9 = 0.555… is rational.

Worked example

Example 2: Square roots

Is √50 rational or irrational?

7² = 49 and 8² = 64, so 50 is not a perfect square.
So √50 is not a whole number, and it is not a fraction either. A calculator shows 7.071 067 8…, which never ends or repeats.

√50 is irrational.

Worked example

Example 3: A decimal with a pattern

Is 0.101 001 000 1… (one more 0 each time) rational or irrational?

There is a pattern, but the gaps keep getting longer.
No block of digits repeats over and over in the same way, and the decimal never ends.

Irrational. Having a pattern is not the same as repeating.

Watch out!

π is not 3.14 and π is not 22/7. Those are approximations.
A calculator rounds. 2 ÷ 3 shows 0.666 666 67 on some screens, but the 6s really repeat forever. A calculator screen alone can't tell you if a decimal repeats; use what you know about the number.

Just for fun: a day for π

Pi Day is celebrated on March 14 (3/14), because π ≈ 3.14. March 14 also happens to be Albert Einstein's birthday. Pie for π, and cake for Einstein? Sounds like a well-rounded day.

Hands-on activity

Measuring π and √2

You need: three or more round objects (a can, a plate, a roll of tape), string, a ruler or measuring tape, a calculator, a square piece of paper (a sticky note works).

Part 1: π. Wrap string once around each round object, then measure the string: that is the circumference C. Measure straight across the middle: that is the diameter d. Find C ÷ d for each object.

Part 2: √2. Measure a side of the square paper. Then fold it corner to corner and measure the diagonal. Find diagonal ÷ side.

What to look for: C ÷ d should come out a little more than 3, near 3.1 or 3.2 every time. Diagonal ÷ side should be near 1.4. Your answers won't be exactly π or √2, because measurements are rounded and these numbers are irrational: no measurement can give every digit.

Math talk: "Why do different-sized circles all give about the same answer?"

Your turn

Rational or irrational: 0.625?

Rational or irrational: 0.252 525…?

Rational or irrational: √15?

Rational or irrational: √121?

Rational or irrational: 2.030 030 003… (one more 0 each time)?

Is 3.14 equal to π? Explain.

Use a calculator: what is 4 ÷ 7? Is 4/7 rational?

More practice: warm-up

Does 0.5 terminate or repeat?

What is the decimal for 1/3?

Is √9 rational?

Is √3 rational?

Round π to two decimal places.

More practice: on your own

Write 7/11 as a decimal. Does it terminate or repeat?

Write 5/12 as a decimal. Which digit repeats?

Which of these are irrational: √20, √25, √0.04, √30?

Find 1.414². Is 1.414 exactly √2?

Between which two whole numbers are π and √10? Which is greater?

Challenge

Explain why the decimal of a fraction like 3/7 must repeat or terminate.

Is √2 + 1 rational or irrational? Explain.

Which is greater, 22/7 or π? By about how much?

Check your understanding

Quiz

Which number is irrational?

The decimal form of every rational number:

Which statement about π is true?

0.101 001 000 1… (one more 0 each time) is:

Which number is rational?

A square has sides of 1 m. How long is its diagonal?

For parents and teachers

Big idea: Under B1.2, students learn that rational numbers have terminating or repeating decimals, while irrational numbers (like √2 and π) have decimals that never end and never repeat. This builds on Grade 7 work with terminating and repeating decimals.

Common difficulties:

Believing π = 3.14 or 22/7. Emphasize "approximation".
Trusting the calculator screen. A calculator shows only so many digits and rounds the last one.
Confusing a pattern with repeating. 0.101 001 000 1… has a pattern but is irrational.

How to help: Do the measuring activity together. Talk about why measurements always come out a bit different. If your child is curious, look up how many digits of π have been calculated; it is a good conversation about how mathematicians know something is true for every digit without checking them all (they use proof).