Order of operations: brackets, powers, roots and BIDMAS

Three shirts at six pounds each. Four pounds for delivery. Write that as one line of maths and it comes out as four plus six times three. There is an add on that line, and there is a times. Something has to decide which one happens first, because the two orders do not give the same money. Add first. Four and six is ten. Ten times three is thirty pounds. Now times first. Six times three is eighteen. Eighteen plus four is twenty-two pounds. Thirty, or twenty-two. Eight pounds apart, off one line, and nobody has made a single arithmetic mistake.

Maths cannot leave that hanging. One written line has to mean one number, everywhere, for everybody. So the order got settled - and it was settled for a reason you can actually see. Look at what six times three is doing in that line. It is not a step. It is a bundle. It is three shirts, priced up, packed into one number: eighteen pounds. You cannot add the delivery onto the shirts until you know what the shirts cost. The bundle has to be worked out before it can join anything. That is the whole reason multiplying goes ahead of adding. Dividing is the same job in reverse. It cuts something into a share, and a share is a bundle too. So multiply and divide both go ahead of add and subtract - not because a rule says so, but because they build the pieces that adding and subtracting move around. Try one on that. Two plus five times four. I am not after a method here. I only want to know which of these two is right: twenty-eight, or twenty-two? Have a think. I'll wait. The answer is twenty-two. Five times four is a bundle worth twenty, and the two joins on afterwards. Twenty-eight comes from reading straight across, and straight across is the one reading the world does not use.

Now the two you have not met yet. Both of them go in front of multiplying. Brackets first. Brackets are not really maths - they are an instruction from whoever wrote the line. They say: this bit is one thing, settle it before you do anything else with it. An instruction that direct outranks everything. Then powers and roots. A power is multiplying packed tighter still. Five squared is five times five, sealed shut into one value, so it settles before the ordinary multiplying around it. A root sign does a bracket's job too: everything underneath it is one thing, worked out first. The one-over key sits in that same place - one divided by your number, worked out as a single value. Powers, roots and reciprocals all seal something shut, so all three settle second. So operations do not go in the order they are written down. They queue up by type, and there are four places in the queue. Brackets. Powers and roots. Multiply and divide. Add and subtract. Where an operation sits in that queue decides the answer. You may already have a word for this: BIDMAS. B for brackets, I for indices, which is the posh word for powers, then divide, multiply, add, subtract. It is a genuine help. It also mislabels two things, and both of them matter. First, BIDMAS puts D in front of M, as though dividing outranks multiplying. It does not. They share a place. Take twelve divided by two, times three. Read left to right, that is six times three, which is eighteen. Let the times jump ahead and you get two. Eighteen is right. Second, A in front of S, as though adding outranks subtracting. It does not. Ten minus four plus three. Do the adding first and you get ten minus seven, which is three - and three is wrong. They share a place, so you go left to right. Ten minus four is six, plus three is nine. So keep the letters if they help, and staple this to them. Four places in the queue, and a tie goes left to right.

Paper first, keys second. Here is one with three of the four places in it. Work out bracket, seven minus two, close bracket, all squared, then divided by five. Bracket first, because brackets always are. Seven minus two is five. The bracket has done its job now, so it goes away, leaving five squared divided by five. Powers next. Five squared is five times five, which is twenty-five. The line is now twenty-five divided by five. That leaves a single divide. Twenty-five divided by five is five. Five is exact - a whole number, nothing left over - so you write five and stop. No decimal, no rounding, nothing after it. Notice the arithmetic inside each step is the ordinary kind. The video on adding, subtracting, multiplying and dividing positive and negative integers is where that gets taught. This video is only ever about the order you do it in.

Now the keypad. Knowing the queue and typing the queue are two different skills, and the second one is where this comes apart. Examiners wrote this about a calculator paper. Some students who did not understand BIDMAS and thus entered the values and operations into their calculator in the wrong order came up with predictably incorrect answers. Here is the useful part. A scientific calculator already knows the queue - it was built with the queue inside it. Type the line exactly as it is written, brackets and all, then press equals once, at the very end. The first way people lose that is pressing equals partway through. Four plus six, equals, gives ten. Then times three gives thirty. Equals does not mean and now the next bit. It means this calculation finishes here. You have told it the line ended early. The second way is leaving brackets out because they look like decoration. Type seven minus two, squared, divided by five with no brackets, and the squaring lands on the two alone. Four divided by five is nought point eight. Seven minus nought point eight is six point two. Right machine, wrong line. So press it in this order. Open bracket, seven, minus, two, close bracket. Then the squared key. Then divide. Then five. Then equals, once. Five. One more to watch for. If that division is printed as a fraction, with the bracket and the square sitting on top of a five, the fraction bar is quietly doing a bracket's job. Type brackets round the top, and round the bottom as well.

One for you now. You have every piece of this already. Work out three plus four times bracket, nine minus six, close bracket. Say which operation you do first, then finish the whole thing. Pause here and work it through. I'll wait. First job is the bracket, because it is at the front of the queue. Nine minus six is three. The line is now three plus four times three. Then multiply and divide come ahead of add and subtract, so four times three is twelve. Last of all, three plus twelve is fifteen. Fifteen, exactly. If you got that, you have it. If you got twenty-one, you did the bracket and then went straight across: three plus four is seven, seven times three is twenty-one. Same numbers, wrong queue.

Time to make this small enough to carry. Four places in the queue, and a tie goes left to right. Brackets first, because somebody put them there on purpose. Powers and roots second, because they seal a value shut. Multiply and divide third, because they build the bundles. Add and subtract last, because they move finished bundles around. And when two operations share a place, neither one jumps. Left to right settles it - which is what saves you on ten minus four plus three. On a calculator: type the line as written, keep every bracket, equals once at the end.

Next in the chapter: Inverse operations - running a calculation backwards to find the number you started from.

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