“You can use BIDMAS,” said the student, and the Mathematical Ninja gave him a piece of paper and a marker.

“Write BIDMAS on here, really big.” While the student complied, the Mathematical Ninja fired up the flamethrower. “Hold it up in front of your face.”

“You know, I’d really rather not.”

“It’s ok, I’m a professional.”

Two singed eyebrows and one incinerated piece of paper later, the Mathematical Ninja smiled.

“I take it we don’t use BIDMAS here,” said the student.

“You take it correctly,” said the Mathematical Ninja, and drew this on the board:

B

P

R

M

D

A

S

“B for brackets. Powers and roots?”

“Exactly,” said the Mathematical Ninja. “As far as A-level goes, this is probably the most useful table you’ll see.”

“More than the normal distribution?”

“Yes. That will only get you a few marks.”

“Wow.”

“Wow, indeed. It has four main uses, and one nice explanation.”

1. Order of operations

“BIDMAS implies - unless it’s taught really carefully, that division comes before multiplication.”

“It doesn’t?”

“No, it happens at the same time as multiplication, and it doesn’t matter what order you do it in. They’re different sides of the same coin - you can even take the extremist view that dividing is really multiplying by a fraction.”

“I’d rather not.”

“Quite right, too. Although it can be a helpful viewpoint to know about.”

“So, if I have a complicated expression to work out, I do the brackets first, then the powers and roots at the same time, then multiply and divide at the same time, then add and subtract at the same time.”

“Correct.”

2. Power laws

“Quick, what’s $10^3 \times 10^2$?”

“$10^5$!”

“So, when you multiply things with the same base, you add the powers.”

“Oh! It’s the line below.”

“Exactly! So, dividing things with the same base, you subtract the powers; powering a power becomes a multiply…”

The student nodded. “Very nice.”

3. Laws of logarithms

“I hate logarithms.”

“Don’t make me tap out rhythms using logs on your knees, then you really will hate them. Alternatively, you can use this table thing here.”

“Let’s see… oh! $\log(ab) = \log(a)+\log(b)$ - so a multiply in the bracket becomes a plus outside!”

“That’s it,” said the Mathematical Ninja. “Moving out of brackets, you move away from the B; moving into brackets, you move towards the B.”

“I see. That’s for another post, though, isn’t it?”

“It is.”

The explanation

“So, why does it work?”

“I’m glad you asked that,” said the Mathematical Ninja. “It’s all to do with repeated operations. ‘Multiply’ just means ‘add something to itself a certain number of times’ - so multiplying is a level up from adding.”

“Aha! And powers are repeated multiplying, so we get A, M and P up the way.”

“And their inverse operations - subtraction, division and rooting - are right beside them.”

“While brackets are only there to keep things tidy.”

The Mathematical Ninja nodded, and reminisced fondly about Boodles, which was a Posh Restaurant that Made Delicious Apple Sauce.