Rational and Irrational Numbers
Every real number is rational or irrational. A rational number can be written as a fraction. An irrational number can't.
What you'll be able to do
- Sort numbers into rational and irrational.
- Write a repeating decimal as a fraction.
- Tell which fractions give a decimal that ends.
- Know which sets a number is in.
The big idea
Two kinds. A rational number is a fraction of two integers, like 3/4 or −5/1. An irrational number is not.
Look at the decimal. A rational number's decimal ends, like 0.375. Or a block of digits repeats, like 0.454545….
Irrational decimals. They never end and never repeat a block. √2 = 1.41421356… and π = 3.14159… are like this.
Roots. √49 = 7, so it is rational. 50 is not a perfect square, so √50 is irrational.
Repeats to fractions. Call the number x. Times it by 10 or 100 so the repeats line up. Then take one from the other.
An example. x = 0.272727…. Then 100x = 27.272727…. Subtract: 99x = 27. So x = 27/99 = 3/11.
When the repeat starts late. For 0.8333…, line up 10x and 100x. They both end in 333…. So 90x = 75.
Which decimals end? Write the fraction in lowest terms. If the bottom has only 2s and 5s, the decimal ends.
Number sets. Whole numbers are 0, 1, 2 and up. Integers add the negatives. All of them are rational and real.
Words you need
- Rational number
- A number you can write as a fraction of two integers.
- Irrational number
- A real number that is not rational. Its decimal never ends or repeats.
- Integer
- A whole number or its negative: …, −2, −1, 0, 1, 2, ….
- Whole number
- 0, 1, 2, 3 and so on.
- Real number
- Any number on the number line, rational or irrational.
- Repeating decimal
- A decimal with a block that repeats forever. 0.454545… can also be written 0.4̅5̅, with a bar on the block.
- Perfect square
- A number like 49 that is a whole number times itself: 7 × 7.
- Product
- The answer when you multiply.
Is it rational?
| The number | Rational? | Why |
|---|---|---|
| −5, 0, 12 | Yes | Integers are fractions: −5 = −5/1 |
| 0.375 | Yes | The decimal ends: 3/8 |
| 0.454545… | Yes | A block repeats: 5/11 |
| √49 | Yes | √49 = 7 |
| √50, π | No | The decimal never ends or repeats |
Worked examples
Example 1
Write 0.272727… as a fraction in lowest terms.
- Let x = 0.272727…. Then 100x = 27.272727….
- Subtract: 100x − x = 27. So 99x = 27.
- x = 27/99 = 3/11.
Answer: 3/11
Example 2
Write 0.8333… as a fraction in lowest terms.
- Let x = 0.8333…. Then 10x = 8.333… and 100x = 83.333….
- Subtract: 100x − 10x = 75. So 90x = 75.
- x = 75/90 = 5/6.
Answer: 5/6
Example 3
Sort 3/8, 5/12, √12 and √25 into rational and irrational. For each fraction, does its decimal end?
- √25 = 5, 3/8 and 5/12 are rational. 12 is not a perfect square, so √12 is irrational.
- In lowest terms, a fraction ends when its bottom has only 2s and 5s.
- 8 = 2 × 2 × 2, so 3/8 = 0.375 ends. 12 = 2 × 2 × 3, so 5/12 = 0.41666… repeats.
Answer: Rational: 3/8 = 0.375, 5/12, √25 = 5. Irrational: √12
Common mistakes
Calling every root irrational. √49 = 7. Only roots of numbers that are not perfect squares are irrational.
Calling a repeating decimal irrational. If a block repeats forever, the number is a fraction.
Using 100x − x when only one digit repeats late. Line up the repeats first: 0.1666… needs 100x − 10x.
Trusting a calculator's last digit. A calculator stops. √2 keeps going and never repeats.
Practice: 20 questions
5 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Sort these numbers into rational and irrational: −7, 0.5333…, √36, √10, 4/9, 0.1010010001…, π.
- Which sets hold −12/4: whole numbers, integers, rational numbers, irrational numbers, real numbers?
- Write 0.454545… as a fraction in lowest terms.
- A 1 m ribbon is cut into 11 equal pieces. Write each length as a decimal. Does it end or repeat? Word problem
- Sam says √49 is irrational because it has a root sign. Find his mistake.
Tier 2 Intermediate One more step to think through
- Write 0.1666… as a fraction in lowest terms. Only the 6 repeats.
- Write 2.363636… as an improper fraction in lowest terms.
- A 7 m rope is cut into 12 equal pieces. Kofi says each piece is 0.58 m. Is that exact? Write the exact decimal. Word problem
- Order from least to greatest: 0.45, 0.454545…, 0.4555…, 5/11.
- Which are rational: √0.49, √0.9, √(4/25), ∛(−27)?
Tier 3 Advanced Several steps - find the hidden structure
- Find 0.333… + 0.181818… as a fraction. Then write the sum as a decimal.
- Four stores sell sticker packs. Store A sells 40 for $3. Store B sells 12 for $5. Store C sells 56 for $7. Store D sells 75 for $9. Without long division, which stores have a price per sticker that ends as a decimal? Word problem
- Write 0.0545454… as a fraction in lowest terms. The block 54 repeats.
- A round track has a diameter of 50 m. Is the distance around it a rational number of meters? Word problem
- A square tile is 1 m on each side. Its diagonal is exactly √2 m. Dee's calculator shows 1.414213562 m. She says the diagonal is rational because the decimal ends. Find her mistake. Word problem
Tier 4 Challenge Stretch problems, some above grade level
- Write 0.999… as a fraction. Is 0.999… less than 1?
- Find a rational number and an irrational number between 1.4 and 1.5.
- Jo says two irrational numbers always add to an irrational number. Test 3 + √2 and 5 − √2.
- 1/7 = 0.142857142857…. What is the 100th digit after the decimal point?
- No real number squares to −1. Mathematicians name a new number i, with i² = −1. Is i a real number? Find i⁴. Above grade level
Answers
Tier 1 Foundations
- Rational: −7, 0.5333…, √36 = 6, 4/9. Irrational: √10, 0.1010010001…, π. A rational number can be written as a fraction of integers. 0.5333… repeats, so it is 8/15. 10 is not a perfect square. 0.1010010001… never repeats a block. π never ends or repeats.
- −12/4 = −3. It is an integer, a rational number and a real number. It is not a whole number, because it is below 0. Every rational number is a real number.
- 5/11. Call x = 0.454545… Then 100x − x = 45, so 99x = 45. So x = 45/99 = 5/11.
- 1/11 = 0.090909… The block 09 repeats. It is rational.
- Rational: √49 = 7. A root sign doesn't make a number irrational. 49 is a perfect square. A rational number can be written as a fraction of integers.
Tier 2 Intermediate
- 1/6. Call x = 0.1666… Then 100x − 10x = 15, so 90x = 15. So x = 15/90 = 1/6.
- 26/11. Call x = 0.363636… Then 100x − x = 36, so 99x = 36. So x = 36/99 = 4/11. Add the whole number: 2 + 4/11 = 26/11.
- No. Each piece is 7/12 m = 0.58333… m. 0.58 m is rounded. Only the 3 repeats. It is rational.
- 0.45 < 0.454545… = 5/11 < 0.4555… 0.454545… and 5/11 are the same number, 5/11. Compare digit by digit: 0.45, 0.454545…, 0.4555….
- Rational: √0.49 = 0.7, √(4/25) = 0.4, ∛(−27) = −3. Irrational: √0.9. A rational number can be written as a fraction of integers. √0.9 is irrational.
Tier 3 Advanced
- 17/33 = 0.515151… 0.333… = 1/3 and 0.181818… = 2/11. Add: 1/3 + 2/11 = 17/33.
- Store A, Store C, Store D. In lowest terms, a decimal ends when the bottom has only 2s and 5s. 3/40: 40 = 2 × 2 × 2 × 5, so it ends. 5/12: 12 = 2 × 2 × 3, so it repeats. 7/56 = 1/8: 8 = 2 × 2 × 2, so it ends. 9/75 = 3/25: 25 = 5 × 5, so it ends.
- 3/55. Call x = 0.0545454… Then 1000x − 10x = 54, so 990x = 54. So x = 54/990 = 3/55.
- No. The distance is 50π m, which is irrational. A rational number can be written as a fraction of integers. 50π is a whole number times π, so it is irrational.
- √2 is irrational. The calculator only shows the first 10 digits. The calculator stops after 10 digits. The real decimal of √2 keeps going. A rational number can be written as a fraction of integers. 2 is not a perfect square.
Tier 4 Challenge
- 0.999… = 1. It is not less than 1. Let x = 0.999…. Then 10x − x = 9.999… − 0.999… = 9. So 9x = 9 and x = 1.
- 1.45 (rational) and √2 (irrational). Many answers work. 1.45 is a fraction: 145/100. √2 is between them, because 1.4² = 1.96 and 1.5² = 2.25.
- Rational: (3 + √2) + (5 − √2) = 8. Irrational: 3 + √2, 5 − √2. The √2 parts can cancel. Then the sum is a whole number. 3 + √2 and 5 − √2 are both irrational. Their sum is 3 + 5 + √2 − √2 = 8.
- 8. The block 142857 has 6 digits. After 16 full blocks, 96 digits are used. The 100th is the 4th digit of the block: 8.
- i is not a real number. i⁴ = 1. No real number squares to −1. i⁴ = (i²)² = (−1)² = 1.
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