B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Rational or Irrational? Decimals, Fractions and Repeating Patterns

B1.2 A1About 70 minutes

Learning goals

I can explain why every fraction becomes a terminating or a repeating decimal.
I can predict whether a fraction will terminate or repeat by looking at its denominator.
I can write repeating decimals with bar notation and convert them to fractions using algebra.
I can tell rational and irrational numbers apart from their decimal forms, and explain what happens when I combine them.

Words to know

terminating decimal
A decimal that ends, like 0.375.

repeating decimal
A decimal in which a block of digits repeats forever, like 0.272727…

period
The block of digits that repeats. In 0.272727… the period is 27 (length 2).

bar notation
A line drawn over the repeating block. 0.272727… is written as 0.27 with a bar over 27.

prime factorization
Writing a number as a product of primes, such as 40 = 2 × 2 × 2 × 5.

lowest terms
A fraction whose numerator and denominator have no common factor except 1.

A calculator puzzle

Type 2 ÷ 3 on a calculator. You might see 0.6666666667. Now type √2. You might see 1.414213562. Both show about ten digits, so the screen cannot tell you which is rational! The 2/3 really goes on as 0.6666… forever; the calculator rounded the last digit. √2 goes on forever too, but its digits never fall into a repeating block.

In this lesson you will learn to tell the difference by reasoning, not by trusting a screen.

Why every fraction terminates or repeats

When you divide a by b with long division, each step leaves a remainder smaller than b. For 1 ÷ 7 the only possible remainders are 0, 1, 2, 3, 4, 5 and 6.

If you ever get remainder 0, the division stops: the decimal terminates.
If not, after at most b − 1 steps a remainder must come back. From then on, the same steps happen again, so the digits repeat.

So every rational number is a terminating or repeating decimal. And it works backwards too: every terminating or repeating decimal can be written as a fraction. That gives a test: a decimal that never ends and never repeats is irrational.

Long division of 1 ÷ 7. The remainders (in circles) go 1 → 3 → 2 → 6 → 4 → 5 and then back to 1, so the digits (on the arrows) 1, 4, 2, 8, 5, 7 repeat forever: 1/7 = 0.142857142857…

Bar notation puts a line over the block that repeats. In 1/6 = 0.1666…, only the 6 repeats, so the bar goes over the 6 alone.

Terminate or repeat? Look at the denominator

Write the fraction in lowest terms, then find the prime factors of the denominator.

If the only prime factors are 2 and/or 5, the decimal terminates. (That is because 10 = 2 × 5, so the fraction can be rewritten over 10, 100, 1000, ...)
If any other prime appears (3, 7, 11, ...), the decimal repeats.

Worked example

Example 1: Predict without dividing

Will each fraction terminate or repeat? 7/40, 5/12, 9/15

7/40: 40 = 2 × 2 × 2 × 5. Only 2s and 5s, so it terminates. (7/40 = 175/1000 = 0.175.)
5/12: 12 = 2 × 2 × 3. There is a 3, so it repeats. (5/12 = 0.41666…)
9/15: first reduce! 9/15 = 3/5. The denominator 5 has only 5s, so it terminates: 0.6.

7/40 terminates, 5/12 repeats, 9/15 terminates. Always reduce first.

Worked example

Example 2: A one-digit repeat to a fraction

Write 0.777… as a fraction.

Let x = 0.777…
The period has 1 digit, so multiply by 10: 10x = 7.777…
Subtract: 10x − x = 7.777… − 0.777…, so 9x = 7. The endless tails cancel.
x = 7/9.

0.777… = 7/9

Worked example

Example 3: A two-digit repeat

Write 0.454545… as a fraction in lowest terms.

Let x = 0.454545… The period (45) has 2 digits, so multiply by 100: 100x = 45.454545…
Subtract: 100x − x = 45, so 99x = 45.
x = 45/99 = 5/11 (divide top and bottom by 9).

5/11. Check: 5 ÷ 11 = 0.4545… ✓

Worked example

Example 4: The repeat starts later

Write 2.1333… as a fraction.

Let x = 2.1333… Move the decimal so the repeating part starts right after it: 10x = 21.333…
Move one more period: 100x = 213.333…
Subtract: 100x − 10x = 213.333… − 21.333…, so 90x = 192.
x = 192/90 = 32/15 (divide by 6).

32/15 (or 2 2/15). Check: 32 ÷ 15 = 2.1333… ✓

The shortcut you can explain

For a repeat that starts right after the decimal point, the fraction is the period over the same number of 9s: 0.777… = 7/9, 0.4545… = 45/99, 0.123123… = 123/999. Example 3 shows why: multiplying by 10ⁿ and subtracting leaves (10ⁿ − 1)x, and 10ⁿ − 1 is a string of n nines.

Combining rational and irrational numbers

rational + irrational = irrational, always. (If √2 + 3 were a fraction q, then √2 = q − 3 would be a fraction too, which is impossible.) The same goes for subtracting, and for multiplying an irrational by a non-zero rational: 5√2 and π/4 are irrational.
irrational + irrational can go either way: √2 + √3 is irrational, but √2 + (−√2) = 0.
Rational approximations are not the number. 22/7 = 3.142857142857… repeats, but π = 3.14159265… does not. They agree only to two decimal places.

Hands-on activity

Fraction detective with a calculator

You need: a calculator (a phone calculator is fine), paper and a pencil.

Make a table with columns: fraction, denominator's prime factors, prediction (terminate or repeat), calculator result.
Test these: 3/8, 2/9, 7/20, 1/12, 5/16, 4/13, 6/25, 11/30.
Predict before you divide. Then check.
For each repeating one, mark the period with a bar.

Model answers: 3/8 = 0.375 (T), 2/9 = 0.222… (R), 7/20 = 0.35 (T), 1/12 = 0.08333… (R, only the 3 repeats), 5/16 = 0.3125 (T), 4/13 = 0.307692307692… (R, period 307692), 6/25 = 0.24 (T), 11/30 = 0.3666… (R, only the 6 repeats).

Bonus: Why does the calculator show 0.3076923077 for 4/13? (It rounds the last digit it can show.)

Just for fun

A funny-but-true fact: 1/7 = 0.142857…, and 2/7, 3/7, 4/7, 5/7 and 6/7 use the same six digits in the same circular order, just starting at a different place. For example, 2/7 = 0.285714… Seven is a show-off.

Your turn

Predict whether 11/40 terminates or repeats, then find its decimal.

Predict whether 7/18 terminates or repeats, then find its decimal.

Write 0.555… as a fraction.

Write 0.818181… as a fraction in lowest terms.

Write 0.8333… as a fraction.

Is 0.12345678910111213… (the counting numbers written one after another) rational or irrational?

Is 3 + √5 rational or irrational? Explain.

More practice: warm-up

Write 3/4 as a decimal. Does it terminate?

Write 2/3 as a decimal using bar notation.

What is the period of 0.181818…?

Is 0.25 rational?

Write 0.333… as a fraction.

More practice: on your own

Without dividing, decide which of 3/16, 5/24, 7/35 terminate.

Write 0.243243243… as a fraction in lowest terms.

Write 1.0666… as a fraction.

Find two irrational numbers whose product is rational.

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

Challenge

Show that 0.4999… (with the 9 repeating) is equal to 1/2.

1/7 has a period of 6 digits. What is the longest period any fraction with denominator 13 could have? Explain using remainders.

Is the sum of a repeating decimal and a terminating decimal rational? Explain.

Check your understanding

Quiz

Which fraction gives a terminating decimal?

What is 0.636363… as a fraction in lowest terms?

Which number is irrational?

Why must a fraction's decimal repeat if it does not terminate?

Which value is irrational?

Where should the bar go in 0.58333…?

For parents and teachers

Big idea: This lesson finishes B1.2 by showing how rational and irrational numbers are defined through their decimals: rational numbers terminate or repeat, irrational numbers do neither. The algebra method (multiply by 10ⁿ and subtract) is students' first taste of using a variable to prove something, which they will use again in Grade 10.

Common difficulties:

Not reducing first. 9/15 looks like it should repeat (15 = 3 × 5), but 9/15 = 3/5 terminates.
Putting the bar over too many digits (0.1666… is not 0.16 repeating).
Multiplying by the wrong power of 10 when the repeat starts later. Encourage two equations whose decimal tails match exactly before subtracting.
Trusting the calculator display. Ten digits cannot show whether a decimal repeats forever.

How to help: Ask your teen to show you why 0.999… = 1 using the algebra method (it surprises most adults!), and to explain it in their own words.