Rational, irrational, and scientific notation
What this lesson covers
- Non-repeating, infinite decimals are irrational numbers; some of them are represented by special symbols, such as √2 and π.
- The terms index, power, and exponent are used interchangeably.
- For the number aⁿ, a represents the base, and n represents the exponent.
One question, asked of every number
- Can this number be written as a fraction of two whole numbers?
- Yes, and it is rational. No, and it is irrational.
- That one question sorts everything in this lesson.
Fractions, decimals and percentages are one number
- Three eighths, nought point three seven five, 37.5 percent.
- Nothing changed between those. Only the costume did.
- Divide for the decimal, multiply by a hundred for the percent.
| Fraction | Decimal | Percentage |
|---|---|---|
| 3/8 | 0.375 | 37.5% |
| 2/5 | 0.4 | 40% |
| 7/20 | 0.35 | 35% |
| 1/3 | 0.333… | 33.3% |
Divide it out and watch what happens
- A quarter stops dead at nought point two five.
- A third never stops, but it repeats forever.
- Every fraction does one or the other. There is no third option.
| Fraction | Dividing gives | So it is |
|---|---|---|
| 1/4 | 0.25, it stops | rational |
| 1/3 | 0.333…, it repeats | rational |
| 5/11 | 0.4545…, it repeats | rational |
| √2 | 1.41421…, neither | irrational |
The numbers that do neither
- The square root of two runs forever with no repeating block.
- So does pi. So does the cube root of ten.
- No fraction of whole numbers will ever equal them.
| Number | Its decimal |
|---|---|
| √2 | 1.414213562… no pattern |
| π | 3.141592653… no pattern |
| ³√10 | 2.154434690… no pattern |
| As a fraction? | None of them can be |
Telling them apart in practice
- A root sign does not make a number irrational.
- The square root of 25 is 5, and 5 is a whole number.
- 22 over 7 looks like pi and is an ordinary fraction.
| Number | Written out | Which is it |
|---|---|---|
| √25 | exactly 5 | rational |
| √26 | 5.0990195… | irrational |
| 22/7 | 3.142857142857… | rational |
| π | 3.14159265… | irrational |
Putting a mixed bag in order
- A third, three eighths, 38 percent and two fifths.
- You cannot compare those as they are written.
- Turn them all into decimals first, then it is easy.
| Number | As a decimal | Order |
|---|---|---|
| 1/3 | 0.3333… | smallest |
| 3/8 | 0.375 | second |
| 38% | 0.38 | third |
| 2/5 | 0.4 | largest |
Base, exponent, and three words for the same thing
- In two to the fifth, the 2 is the base.
- The 5 is the exponent: how many twos are multiplied.
- Index, power and exponent all mean that same 5.
| In 2⁵ | Its name | What it does |
|---|---|---|
| 2 | the base | the number multiplied |
| 5 | the exponent | how many of them |
| 2⁵ | the power | 2 × 2 × 2 × 2 × 2 = 32 |
| Index, power, exponent | one idea | three words for n |
Scientific notation, and its one rule
- Every number gets written as something times a power of ten.
- That something has to be at least 1 and under 10.
- The power of ten says how far the decimal point moved.
| The form | What it means |
|---|---|
| Written a × 10ᵏ | a is at least 1 and under 10 |
| k | how far the point moves |
| Numbers above 10 | k is positive |
| Numbers under 1 | k is negative |
Writing a big number down
- 384 400 kilometres to the moon. Move the point left.
- Five places, landing on 3.844.
- So it is 3.844 times ten to the fifth.
| Number | Point moves | Written as |
|---|---|---|
| 384 400 | 5 places left | 3.844 × 10⁵ |
| 93 000 000 | 7 places left | 9.3 × 10⁷ |
| 1 500 | 3 places left | 1.5 × 10³ |
And a very small one
- Same move, other direction, so the exponent goes negative.
- Nought point nought nought nought nought nought two five.
- Six places right, so 2.5 times ten to the minus six.
| Number | Point moves | Written as |
|---|---|---|
| 0.000 002 5 | 6 places right | 2.5 × 10⁻⁶ |
| 0.007 | 3 places right | 7 × 10⁻³ |
| 0.000 000 1 | 7 places right | 1 × 10⁻⁷ |
Which one is bigger?
- Look at the exponent before you look at anything else.
- Ten cubed beats ten squared whatever the front number is.
- Only when the exponents match does the front number decide.
| Number | Exponent | Really |
|---|---|---|
| 4.2 × 10³ | 3 | 4 200 |
| 1.1 × 10³ | 3 | 1 100 |
| 9.8 × 10² | 2 | 980 |
| Exponent first | then a | 4.2 × 10³ wins |
Multiplying and dividing them
- Front numbers with front numbers, powers with powers.
- Multiplying tens adds the exponents. Dividing subtracts them.
- Then check the front number is still under 10.
| Working | Front | Powers |
|---|---|---|
| Multiplying | 3 × 2 = 6 | 10⁴ × 10⁵ = 10⁹ |
| Dividing | 8 ÷ 4 = 2 | 10⁷ ÷ 10³ = 10⁴ |
| Another one | 5 × 4 = 20 | gives 20 × 10⁹ |
| Not finished | 20 is over 10 | so 2 × 10¹⁰ |
Why anyone bothers
- A calculator that shows 3.2E9 is speaking this language.
- It also makes the size obvious at a glance.
- Ten to the ninth is a billion, whatever the front number is.
What to take away
- Six things, and the first three are one idea.
| Rational | it can be written as a fraction |
| Its decimal | always stops or repeats |
| Irrational | never stops and never repeats |
| √2 and π are irrational | 22/7 is not, it repeats |
| In aⁿ | a is the base, n is the exponent |
| Scientific notation | a × 10ᵏ, with a under 10 |
Your turn: four questions
- Pause here and write all four out properly.
- Question 2 has one irrational number in it, not two.
- Question 4 is not finished when you get the first answer.
| Question | |
|---|---|
| 1 | Write 0.45 as a fraction and a percent. |
| 2 | Which is irrational: √25, √26, 22/7? |
| 3 | Write 0.000 084 in scientific notation. |
| 4 | Work out (6 × 10⁵) × (5 × 10³). |
Show answer
1. 0.45 is 45 hundredths, so 45/100, which cancels by 5 to 9/20. As a percent it is 45%, because a percent is already a fraction of a hundred. 2. The square root of 26 is the irrational one. The square root of 25 is exactly 5, and 22/7 is a fraction, so both of those are rational however untidy 22/7 looks written out. 3. 8.4 × 10⁻⁵. The point moves five places right to get back to 0.000 084, so the exponent is negative five. 4. 6 times 5 is 30 and 10⁵ times 10³ is 10⁸, giving 30 × 10⁸. That is not finished, because 30 is not under 10: it tidies to 3 × 10⁹.
The answers
- 0.45 is 45 hundredths, so 9 over 20, and 45 percent.
- The square root of 26 is the irrational one.
- 0.000 084 is 8.4 times ten to the minus five.
- 30 times ten to the eighth tidies to 3 times ten to the ninth.
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