Reciprocals, and why decimal rules are fraction rules
What this lesson covers
- Every non-zero number has a multiplicative inverse (reciprocal), and their product is 1 (e.g. 5 and 1/5 are reciprocals, so 5 × 1/5 = 1/5 × 5 = 1).
Two topics that are secretly one topic
- You were taught fraction arithmetic and decimal arithmetic separately.
- They are the same arithmetic. Every decimal is a fraction.
- Which means the decimal rules you half-remember have reasons.
The number that cancels another one out
- Every number except zero has a partner called its reciprocal.
- Multiply the two together and you always get exactly 1.
- Five and a fifth. Two thirds and three halves.
| Number | Reciprocal | Product |
|---|---|---|
| 5 | 1/5 | 5 × 1/5 = 1 |
| 2/3 | 3/2 | 6/6 = 1 |
| 1/4 | 4 | 4/4 = 1 |
| 7/8 | 8/7 | 56/56 = 1 |
| 0 | none | nothing times 0 gives 1 |
Except zero, and it is worth knowing why
- What would you multiply zero by to get 1?
- Nothing works. Zero times anything is zero.
- That is the same reason you cannot divide by zero.
So that is why flipping works
- Last lesson: dividing by 2/3 gave the same answer as times 3/2.
- Dividing by a number undoes multiplying by it.
- The reciprocal is the thing that undoes it, because together they make 1.
Adding: make the pieces the same size
- Three quarters plus one sixth. Quarters and sixths do not line up.
- Rewrite both in twelfths: nine twelfths and two twelfths.
- Now they are the same kind of thing, so just add the tops.
| Step | Why | |
|---|---|---|
| 3/4 + 1/6 | quarters and sixths differ | |
| 9/12 + 2/12 | both in twelfths now | |
| 11/12 | add the tops, keep the bottom |
Subtracting: exactly the same move
- Two thirds minus a quarter. Twelfths again.
- Eight twelfths minus three twelfths.
- Five twelfths. Nothing new to learn here.
| Step | Why | |
|---|---|---|
| 2/3 − 1/4 | thirds and quarters | |
| 8/12 − 3/12 | both in twelfths | |
| 5/12 | subtract the tops |
Multiplying needs no lining up at all
- This is the one people overcomplicate, having just done adding.
- Two fifths times three sevenths. No common denominator needed.
- Straight across the top, straight across the bottom.
| Step | What happens | |
|---|---|---|
| 2/5 × 3/7 | no lining up needed | |
| 2 × 3 = 6 | across the top | |
| 5 × 7 = 35 | across the bottom | |
| 6/35 | done |
Dividing: flip the second one
- Three quarters divided by two fifths.
- Flip the second fraction and multiply instead.
- Fifteen eighths, which is one and seven eighths.
| Step | What happens | |
|---|---|---|
| 3/4 ÷ 2/5 | how many two-fifths fit? | |
| 3/4 × 5/2 | multiply by the reciprocal | |
| 15/8 | across the top, across the bottom | |
| 1 7/8 | as a mixed number |
Now decimals, which are already fractions
- Nought point three is three tenths. Written differently, same number.
- Nought point one five is fifteen hundredths.
- So every decimal calculation is a fraction calculation.
| Decimal | As a fraction | Reading |
|---|---|---|
| 0.3 | 3/10 | tenths |
| 0.15 | 15/100 | hundredths |
| 0.045 | 45/1000 | thousandths |
Where the count-the-places rule comes from
- Nought point three times nought point one five.
- As fractions: three tenths times fifteen hundredths.
- Tenths times hundredths is thousandths. Hence three places.
| Line | What it is | |
|---|---|---|
| 0.3 × 0.15 | the decimal version | |
| 3/10 × 15/100 | the same thing as fractions | |
| 45/1000 | tenths × hundredths = thousandths | |
| 0.045 | 1 place plus 2 places makes 3 |
And why you can move both decimal points
- Nought point eight divided by nought point two.
- Eight tenths divided by two tenths. The tenths cancel.
- Eight divided by two is four. That is the whole trick.
| Line | What it is | |
|---|---|---|
| 0.8 ÷ 0.2 | the decimal version | |
| 8/10 ÷ 2/10 | the same thing as fractions | |
| 8 ÷ 2 | the tenths cancel | |
| 4 | so moving both points works |
Last thing: which of these are the same number?
- Four pairs of expressions, all with minus signs in them.
- Three pairs are equal. One pair is not.
- Work out which before the answer comes up.
Three equal, one not
- Order does not matter for adding or multiplying. Those are fine.
- A minus sign on top or in front of a fraction is the same place.
- But minus three squared depends entirely on the brackets.
| Pair | Values | Verdict |
|---|---|---|
| (−3)² and −3² | 9 and −9 | NOT equal |
| −2 + 3 and 3 + (−2) | 1 and 1 | equal |
| 2 × (−3) and (−2) × 3 | −6 and −6 | equal |
| 2/(−3) and (−2)/3 | −2/3 and −2/3 | equal |
What to take away
- Seven things, and the sixth one is the big idea.
| Reciprocal | the number that multiplies with it to give 1 |
| Zero | is the one number without one |
| Adding fractions | needs a common denominator first |
| Multiplying | does not. Straight across |
| Dividing | is multiplying by the reciprocal |
| Decimal rules | are fraction rules underneath |
| (−3)² and −3² | are different numbers |
Your turn
- Pause here and try all four.
- Answers are in the description.
| Question | |
|---|---|
| 1 | What is the reciprocal of 3/8, and what do they multiply to? |
| 2 | Work out 3/4 ÷ 2/5. |
| 3 | Work out 0.4 × 0.07 without a calculator. |
| 4 | True or false: (−5)² = −5². |
Show answer
1. 8/3, and they multiply to 1. That is what makes it the reciprocal. 2. 15/8, or 1 7/8. Flip the second fraction and multiply. 3. 0.028. Tenths times hundredths gives thousandths, so three decimal places, and 4 × 7 = 28. 4. False. (−5)² is 25 and −5² is −25. The brackets decide what gets squared.
The answers
- Flip it, and the pair multiplies to one.
- Tenths times hundredths is thousandths.
- Minus five squared is 25 or −25, by brackets.
| Answer | |
|---|---|
| 1 | 8/3, and they make 1 |
| 2 | 15/8, or 1 7/8 |
| 3 | 0.028 |
| 4 | False |
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