🍰 Fraction Isles · Number Sense

Rational & Irrational Numbers

Sort the real numbers: every fraction is a decimal that stops or repeats, every other decimal is irrational, and both kinds have a place on the number line.

In short

Rational means "can be written as one whole number over another". Irrational means no such fraction exists, however far you look.
A fraction always gives a decimal that either stops or repeats; a decimal that stops or repeats is always a fraction. The two statements are the same fact read in each direction.
A fraction in lowest terms terminates exactly when its denominator is built only from 2s and 5s, because $10 = 2 x 5$ .
To turn a repeating decimal into a fraction, name it x, multiply until a second line has the same endless tail, and subtract so the tails cancel.
sqrt(n) is rational exactly when n is a perfect square. Trap any irrational number between two rationals by squaring candidates and adjusting.

One line, two kinds of number

Every number you have met so far has a home somewhere on the number line, and the whole line together is called the real numbers. What this skill does is sort those numbers into two families.

A number is rational when it can be written as one whole number over another — a ratio, which is where the word comes from. That family is bigger than it first looks:

7 is rational, because $7 = 7/1$ .
$-3/4$ is rational; so is $22/7$ .
0.75 is rational, because it is $75/100$ .
0.333... is rational too, and the next section shows why.

A number is irrational when no such fraction exists, no matter how large you let the top and bottom grow. sqrt(2) and pi are the famous ones. Irrational numbers are not strange or broken: they sit on the line in exact positions, and the diagonal of $a 1$ by 1 square really is sqrt(2) long. They simply cannot be written as a ratio.

Divers at Decimal Reef read depths to hundredths of a fathom, and every one of those readings is rational. The shelf itself does not stop at hundredths — that is where the other family lives.

Every fraction stops or repeats

A fraction is a division waiting to happen: $3/8$ means 3 divided by 8. Do the long division and only two things can happen.

It stops. $3/8 = 0.375$ , and the remainder reaches $0. A$ decimal that stops is called *terminating*.

It repeats. $1/3 = 0.333..., 5/11 = 0.454545..., 1/6 = 0.1666$ ... The remainder never reaches 0, but there are only so many remainders available, so one of them must come back around — and once a remainder repeats, the digits after it repeat too, forever.

Which one you get is decided entirely by the bottom of the fraction once it is in lowest terms. Ten is $2 x 5$ , so a denominator built only from 2s and 5s can be scaled up to $10, 100$ or 1000 and the decimal stops. Any other factor left in the bottom — $a 3, a 7$ , an 11 — and it repeats.

$3/8$ : the bottom is $2 x 2 x 2$ , so it stops. $3/8 = 0.375$
$7/20$ : the bottom is $2 x 2 x 5$ , so it stops. $7/20 = 0.35$
$5/12$ : the bottom is $2 x 2 x 3$ , and that 3 will not go away. $5/12 = 0.41666$ ...

To write a repeating decimal without an endless row of digits, put brackets around the block that repeats. This lesson and every question in this skill use that one convention:

0.(3) means 0.333...
0.(45) means 0.454545...
0.1(6) means 0.1666... — only the 6 repeats, the 1 does not.

That gives the first half of the test for rationality. Stops or repeats means rational. Never stops and never repeats means irrational.

Turning a repeating decimal back into a fraction

If every fraction gives a stopping or repeating decimal, does every repeating decimal come from a fraction? It does, and there is a trick that produces it every time. The idea is to build two numbers with exactly the same endless tail, then subtract so the tails cancel.

Take $x = 0.(45)$ , which is 0.454545...

Two digits repeat, so multiply by 10 twice — that is, by 100.
$100x = 45.454545$ ...
$x = 0.454545$ ...
Subtract: 100x - $x = 45.454545$ ... - $0.454545... = 45$ exactly. The tails were identical, so they vanished.
$99x = 45$ , so $x = 45/99 = 5/11$ .

The subtraction is the whole trick, and it is worth seeing why the denominator comes out as 99: you multiplied by 100 and took one x away, leaving 100 - 1.

When some digits sit before the repeat, clear them first with an extra multiplication. For $x = 0.1(6)$ , multiply by 10 to get $10x = 1.666$ ..., then by 10 again to get $100x = 16.666$ ... Subtracting those two leaves $90x = 15$ , so $x = 15/90 = 1/6$ . Always subtract the two lines whose tails match, not any two lines.

Roots and pi: the other family

The most common irrational numbers you will meet are square roots. sqrt(n) is a whole number exactly when n is a perfect square — $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$ — and irrational for every other whole number.

sqrt(16) $= 4$ , rational. The root sign by itself proves nothing.
sqrt(17) is irrational: 17 sits between 16 and 25, and is neither.

Cube roots behave the same way against the perfect cubes $8, 27, 64, 125$ . And pi is irrational: $22/7$ and 3.14 are near neighbours that make good working values, but neither one $*is* pi$ .

You cannot write an irrational number down exactly as a decimal, so instead you trap it between numbers you can write. To place sqrt(30):

$5 x 5 = 25$ and $6 x 6 = 36$ , and $25 < 30 < 36$ , so sqrt(30) is between 5 and 6.
Now try tenths. $5.4 x 5.4 = 29.16$ , which is under $30. 5.5 x 5.5 = 30.25$ , which is over.
So sqrt(30) is between 5.4 and 5.5, and 30.25 is much nearer 30 than 29.16 is. To one decimal place, sqrt(30) $= 5.5$ .

Squaring your guess is the check that keeps you honest, and it works at every level of precision. Each extra decimal place traps the number in a smaller gap; the gap never closes, which is precisely what being irrational means.

Comparing a mixed bag

Fractions, decimals, roots and pi look nothing alike, so before you can order them they need a common form. Decimals are usually easiest: divide for a fraction, trap a root between tenths, and use 3.14 for pi.

Order $3/8, 0.4$ , sqrt(2) and pi from least to greatest:

$3/8 = 0.375$
$0.4 = 0.4$
sqrt(2) is between 1 and 2, and $1.4 x 1.4 = 1.96$ , so about 1.41
pi is about 3.14

Least to greatest: $3/8, 0.4$ , sqrt(2), pi.

Two warnings are worth carrying. First, the digits printed on the page are not the value: sqrt(11) is about 3.3, nowhere near eleven, and 0.(3) is a third, not a shade above zero. Second, more digits does not mean bigger — 0.45 is smaller than 0.5, however long it looks.

Once every number is a decimal you can also place them all on one number line, which is the clearest picture of what this whole skill is about: rational and irrational numbers share a single line, packed in among each other, with no gap between the two families anywhere along it.

Worked examples

Example 1

Write 0.(45) as a fraction in simplest form.

Give the number a name: let $x = 0.(45)$ , so $x = 0.454545$ ...
The repeating block is 2 digits long, so multiply by $100: 100x = 45.454545$ ...
Line the two up and subtract. The endless tails are identical, so they cancel: 100x - $x = 45.454545$ ... - $0.454545... = 45$ .
$99x = 45$ , so $x = 45/99$ .
Simplify by dividing top and bottom by $9: 45/99 = 5/11$ .
Check: 5 divided by 11 is 0.454545..., which is where we started.

Example 2

Write $7/12$ as a decimal, and say whether $7/12$ is rational or irrational.

In lowest terms the bottom is $12 = 2 x 2 x 3$ . The 3 is not $a 2$ or $a 5$ , so this decimal will repeat rather than stop.
Divide 7 by $12: 7.000 / 12$ gives 0.58, with a remainder that keeps producing 3s.
$7/12 = 0.583333$ ..., which in bracket notation is 0.58(3) — the 5 and the 8 happen once, only the 3 repeats.
It is rational: it was already written as one whole number over another, and its decimal repeats, which is the same conclusion reached twice.

Example 3

Which is larger, sqrt(20) or 4.5? Give sqrt(20) to one decimal place.

Trap sqrt(20) between whole numbers: $4 x 4 = 16$ and $5 x 5 = 25$ , and $16 < 20 < 25$ , so sqrt(20) is between 4 and 5.
Test the halfway point: $4.5 x 4.5 = 20.25$ , which is more than 20. So sqrt(20) is below 4.5, and 4.5 is the larger of the two.
Now narrow it to a tenth: $4.4 x 4.4 = 19.36$ , under $20; 4.5 x 4.5 = 20.25$ , over 20. So sqrt(20) is between 4.4 and 4.5.
20 is nearer 20.25 than 19.36, so to one decimal place sqrt(20) $= 4.5$ .
Both facts hold at once: rounded to one decimal place sqrt(20) shows as 4.5, and its exact value is still a little under 4.5. Rounding never changes which number is larger.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Is this number rational or irrational? 0.24

rational
irrational

Answer: A. rational

The question is whether 0.24 can be written as one whole number over another.
0.24 stops after 2 decimal places, so it is exactly $24/100 = 6/25$ .
So it is rational.

Problem 2

Difficulty 3 of 5

Write $3/16$ as a decimal.

Answer: 0.1875

$3/16$ means 3 divided by 16.
Divide: $3.000 / 16 = 0.1875$ , and the remainder reaches 0, so the decimal stops.
Check by scaling: $3/16 = 1875/10000$ .

Problem 3

Difficulty 4 of 5

Write 0.7(50) as a fraction in simplest form. (The brackets mark the block of digits that repeats forever, so 0.(3) means 0.333... This number is 0.75050...)

Answer: $743/990$

Let $x = 0.7(50)$ , which written out is 0.75050...
Multiply by 10 to walk the digits before the repeat past the point: $10x = 7.5050$ ...
Multiply by 1000 so the repeating tail lines up exactly: $1000x = 750.5050$ ...
Subtract the two lines. The endless tails are identical, so they cancel: $990x = 750$ - $7 = 743$ .
$x = 743/990$ , which is already in lowest terms.

Common mistakes

Calling sqrt(17) rational because it has a root sign that "works out". The root of a number that is not a perfect square never stops and never repeats.
Treating $22/7$ or 3.14 as pi. They are useful neighbours of pi, and both of them are rational — pi itself is not.
Calling a repeating decimal irrational. 0.(3) never stops, but it does repeat, and repeating decimals are all fractions.
Writing 0.(45) as $45/100$ . That denominator only works for a decimal that stops; a repeating block gives a denominator built from 9s, so $0.(45) = 45/99$ .
Ordering numbers by the digits on show: reading sqrt(11) as eleven-ish, or judging 0.45 to be larger than 0.5 because it has more digits.
Rounding an irrational number and then treating the rounded value as exact. sqrt(20) shows as 4.5 to one decimal place, but it is still under 4.5.

What you should be able to do

Decide whether a number is rational or irrational and explain why.
Convert a fraction to a terminating or repeating decimal, and a repeating decimal back to a fraction.
Approximate an irrational number such as sqrt(10) or pi and place it between two rationals.
Order a mixed set of fractions, decimals, roots and pi from least to greatest.

Where this fits in the curriculum

Common Core

8.NS.A.1
Grade 8 — Know that numbers that are not rational are irrational, and convert a decimal expansion that repeats eventually into a rational number.
8.NS.A.2
Grade 8 — Use rational approximations of irrational numbers to compare them, locate them on a number line and estimate expressions.
7.NS.A.2.D
Grade 7 — Convert a rational number to a decimal using long division, and know that the decimal terminates in 0s or eventually repeats.

Ontario

G8.B1.2
Grade 8 — Describe, compare and order numbers in the real number system (rational and irrational numbers), separately and in combination, in various contexts.