Order of operations, signs and reciprocals
What this lesson covers
- The order of operations is important when evaluating or forming expressions. Operations are done as follows: grouped operations (e.g. expressions under a square root, involving the numerator of a fraction, or inside brackets) exponents or powers multiplication and division, from left to right addition and subtraction, from left to right.
- A mnemonic can be used to remember the order of operations, such as GEMA: Grouped (e.g. √(3²+ 4²)), Exponents (e.g. (−2)³), Multiplicative (× and ÷), Additive (+ and −).
- 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).
Why there is an order at all
- Two people work out the same expression and disagree.
- That cannot happen, or written maths would not work.
- So the order is a shared agreement about what binds tightest.
Four tiers, in order
- Grouped things first. Then exponents.
- Then multiply and divide, working left to right.
- Then add and subtract, left to right. GEMA.
| Tier | What it covers | |
|---|---|---|
| G | Grouped | brackets, fractions, roots |
| E | Exponents | powers and roots |
| M | Multiplicative | × and ÷, left to right |
| A | Additive | + and −, left to right |
Grouped means more than brackets
- A root sign groups everything underneath it.
- A fraction bar groups the whole top and the whole bottom.
- That is why the G is grouped and not brackets.
| Expression | What groups | Value |
|---|---|---|
| (8 − 3) | ordinary brackets | obvious |
| √(3² + 4²) | the root groups | = 5 |
| (2 + 4) / 3 | the bar groups | = 2 |
| Not just brackets | anything that encloses |
Minus two squared
- Minus two squared is minus four.
- Minus two, in brackets, squared is positive four.
- The bracket decides what the exponent owns.
| Written | What is squared | Value |
|---|---|---|
| −2² | the 2 is squared | −4 |
| (−2)² | the −2 is squared | 4 |
| The bracket | decides what is squared | |
| Same digits | different expressions |
Powers of negative numbers
- Negatives pair off. Each pair makes a positive.
- An even exponent pairs them all, so the answer is positive.
- An odd one leaves a negative over.
| Power | Why | Value |
|---|---|---|
| (−1)⁴ | four negatives pair up | 1 |
| (−1)⁵ | one is left over | −1 |
| (−2)³ | odd, so negative | −8 |
| (−3)² | even, so positive | 9 |
One expression, tier by tier
- Grouped first: eight take three is five.
- Exponent next: five squared is twenty-five.
- Then times and divide, then the addition. Ten.
| Tier | Doing | Leaves |
|---|---|---|
| 5 + 2 × (8 − 3)² ÷ 10 | start | |
| Grouped | 8 − 3 = 5 | 5 + 2 × 5² ÷ 10 |
| Exponents | 5² = 25 | 5 + 2 × 25 ÷ 10 |
| × and ÷, left to right | 50 ÷ 10 = 5 | 5 + 5 = 10 |
Adding and subtracting with signs
- Adding moves right. Subtracting moves left.
- From minus seven, adding three gets you to minus four.
- Subtracting a negative turns round and goes right.
| Working | On the line | Answer |
|---|---|---|
| −7 + 3 | move 3 right | −4 |
| −7 − 3 | move 3 further left | −10 |
| 5 − (−3) | taking away a debt | 8 |
| Subtracting a negative | is adding |
Multiplying and dividing with signs
- Count the minus signs, then look at whether it is even or odd.
- Two negatives multiplied give a positive.
- One negative in a division leaves the answer negative.
| Working | Minus signs | Answer |
|---|---|---|
| −6 × −4 | the signs pair off | 24 |
| −20 ÷ 5 | one negative | −4 |
| −20 ÷ −5 | two negatives | 4 |
| Count the minuses | even is positive | odd is negative |
Fractions and decimals do the same thing
- A negative fraction is a fraction with a sign on it.
- Find a common denominator, then follow the sign rules.
- Decimals are no different. Nothing new to learn.
| Working | Because | Answer |
|---|---|---|
| −3/4 + 1/2 | common denominator | −1/4 |
| −2.5 × 0.4 | one minus, so negative | −1 |
| −0.6 − 0.75 | both going left | −1.35 |
| Nothing new | the rules do not change |
The reciprocal: whatever undoes a multiply
- Five times a fifth is one. They undo each other.
- Two thirds and three halves do the same thing.
- The test is always the same: does it multiply back to 1?
| Number | Its reciprocal | Check |
|---|---|---|
| 5 | 1/5 | 5 × 1/5 = 1 |
| 2/3 | 3/2 | turn it over |
| −4 | −1/4 | the sign stays |
| −0.2 | −5 | still gives 1 |
Which is why you cannot divide by zero
- Every number except one has a reciprocal.
- Nothing multiplied by zero gives 1. Nothing at all.
- So zero has no reciprocal, and nobody decided that.
And why you flip the second fraction
- Dividing by something is multiplying by what undoes it.
- What undoes two fifths is five halves.
- So three quarters divided by two fifths is fifteen eighths.
| Step | Reason | Result |
|---|---|---|
| 3/4 ÷ 2/5 | dividing by 2/5 | |
| Dividing undoes | so multiply by 5/2 | |
| 3/4 × 5/2 | across the top and bottom | 15/8 |
| Which is | 1 and 7/8 |
One habit worth building
- Write the expression again after every tier.
- It takes four seconds and stops the sign errors.
- Most lost marks here are not understanding. They are rushing.
What to take away
- Seven lines, and the last two are the same idea.
| Grouped, exponents | then × ÷, then + − |
| Grouped is wider than brackets | roots and fraction bars too |
| −2² is −4 | and (−2)² is 4 |
| Even power of a negative | is positive |
| Subtracting a negative | is adding |
| A reciprocal | multiplies back to 1 |
| Dividing by a fraction | multiplies by its reciprocal |
Your turn: four questions
- Pause here and work all four out on paper.
- Question 2 is two different expressions, not one.
- Question 4 wants the check as well as the answer.
| Question | |
|---|---|
| 1 | Work out 12 − 3 × (2 + 2)² ÷ 8. |
| 2 | Evaluate (−3)² and −3². |
| 3 | Work out −5/6 ÷ 2/3. |
| 4 | What is the reciprocal of −0.2? |
Show answer
1. 6. Grouped first: 2 + 2 is 4. Then the exponent: 4² is 16. Then times and divide left to right: 3 × 16 is 48, divided by 8 is 6. Then the subtraction: 12 − 6 = 6. 2. 9 and −9. In the first, the bracket squares the whole negative three. In the second, only the 3 is squared and the minus sign stays outside it. 3. −5/4, or −1.25. Dividing by two thirds means multiplying by three halves: −5/6 × 3/2 = −15/12, which cancels to −5/4. One negative, so the answer is negative. 4. −5. Nought point two is one fifth, so its reciprocal is five, and the negative comes along: −0.2 × −5 = 1, which is the test a reciprocal has to pass.
The answers
- Twelve take six is six.
- Nine, and minus nine. The bracket is the whole difference.
- Minus five sixths times three halves is minus five quarters.
- Minus five, because minus nought point two times minus five is 1.
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