Inverse operations

Someone in your class does the number trick on you. Think of a number, they say. Don't tell me. Double it. Now add ten. Tell me the answer, and only the answer. You say forty-six. They say: you started with eighteen. And they are right, whoever plays and whatever number gets picked. There is no mind-reading in it. They took your answer and walked your two steps backwards. That walk back is the whole of this video. It is what a paper wants when it hands you the end of a calculation and asks for the beginning.

One step first. Undoing is easier to see when there is only one thing to undo. Pick seven, and add six. On the number line you have moved six places further right. You land on thirteen. To get back to seven, you go six places further left. That is subtract six. The two moves cancel each other out, exactly, every time. That is what an inverse operation is. The operation that puts a number back where it started. They come in pairs, and there are only three pairs to hold on to. Add and subtract. Multiply and divide. Square and square root. They work the same way. Square five and you get twenty-five. Square root twenty-five and you are back at five. Here is one for you. A small one. Which of these undoes add six? A: add six again. B: subtract six. C: multiply by six. Pick one of the three. I'll wait. The answer is B, subtract six. Add six moves you six places further right, and subtract six brings you those six places back. A takes you further away. C is a different operation altogether. Keep those three pairs somewhere you can reach. An operation in reverse should be a lookup, not a guess.

Two steps now. And two steps ask a question one step never asks: which do you undo first? Take the trick from the start. Double, then add ten. Eighteen doubled is thirty-six, and ten more makes forty-six. That is the chain running forwards. You have the forty-six. You want the eighteen. So: do you undo the doubling first, or the add ten first? Look at both, then pick. I'll wait. The answer is the add ten. It was the last thing done going forwards, so it is the first thing undone coming back. Here is why. Think about getting dressed. Socks on, then shoes on top. To undo it you take the shoes off first. Not because shoes are special. Because they went on last, so they are the ones on the outside. The add ten went on last, so it is sitting on the outside of forty-six. It comes off first. Forty-six subtract ten is thirty-six. Now the doubling is outermost, so it comes off next. Thirty-six divided by two is eighteen. The number they started with. Last done, first undone. That is your line for this whole video. What decided the order was not which operation it was, or how hard it looked. It was where that operation sat in the chain.

Watch me do one out loud now, with nothing hidden. Think of a number. Subtract nine, then multiply by four. The result is twenty. Work out the number you started with, showing each inverse step. Draw the chain going forwards first, even though you don't know the starting number. An empty box, then subtract nine, then multiply by four, then twenty. The last step in was the multiply by four. So that is the first step out, and its inverse is divide by four. Twenty divided by four is five. So just before the multiplying, the number sitting there was five. One step left, the subtract nine. What undoes it, and what number does that give you? Pause it now and work this one out for yourself. I'll wait. Add nine undoes subtract nine. Five add nine is fourteen. The number thought of was fourteen. Look at what sits waiting there. Undo the multiply by four, get five, stop, and you hand in five. Five is the number halfway through the chain, not the number at the start. Every step gets undone, or you are not back yet. Then check by running forwards. Fourteen subtract nine is five. Five multiplied by four is twenty. That was the result you were given, so fourteen is right.

The next chain carries a minus sign the whole way through the reversal. Keep your eye on it. A number is multiplied by minus three. Then eight is added. The output is minus one. Work backwards to find the original number. Chain forwards: an empty box, multiply by minus three, add eight, and out comes minus one. Add eight went on last, so it comes off first. Its inverse is subtract eight. Minus one, subtract eight. Minus one is already one step below zero. Take eight more away and you are nine steps below zero. Minus nine. One step left. Undo multiply by minus three, and that is divide by minus three. So it is minus nine divided by minus three. A minus divided by a minus gives a positive. Minus nine divided by minus three is three. Not minus three. That sign is the difference between the right answer and a wrong one, and it is why you write it down at every step. If the minus arithmetic itself feels shaky, the video on adding, subtracting, multiplying and dividing positive and negative integers walks through it slowly. So the original number is three. Run it forwards: three multiplied by minus three is minus nine, and minus nine add eight is minus one. That is the output you were handed, so three is right.

Examiners watch this go wrong in two different ways. Here they are, in their own words. The first one. Some students conducted an incorrect inverse operation such as forty-three plus seven, or thirty-six minus four. Right numbers, wrong operation. They added where the undo was a subtract. That is the three pairs doing their job: settle which operation you need before you touch the numbers. The second one. The majority of candidates identified division as the inverse operation needed here. The common error was to ignore the negative and give x equals seven. The method was fine. They chose division, and division was right. The sign got dropped in the middle, and a tidy-looking wrong answer came out. That is the chain with minus three, exactly. On paper that turns into three habits. Write the chain out forwards before you reverse anything. Give every undo step its own line, sign included. Then run it forwards to check. Take a question like this: work backwards to find the number he started with. Those three habits answer it. And the forward check is not politeness. It catches a dropped minus sign while you can still do something about it.

Now a short one, to find out whether this has stuck. A number has six added to it. Then it is divided by two. That gives five. Work backwards and find the number. Pause here and work it through. I'll wait. The answer is four. Divide by two went on last, so undo it first: five multiplied by two is ten. Then undo the add six: ten subtract six is four. Forwards to check: four add six is ten, and ten divided by two is five. It matches. And if you got ten, you undid the dividing and stopped one step short of the start.

Last done, first undone. Five lines now, and the last one closes the chapter. Three pairs carry all of it. Add and subtract. Multiply and divide. Square and square root. Write the chain forwards, start from the answer, and undo every step in reverse order, back to the start. Not only the last one. Keep the sign attached to its number the whole way through. A minus dropped halfway gives you an answer that looks tidy and is not right. Then run it forwards and check it lands on the number you were given. And that is the idea this whole chapter has been circling. A digit's column. A number's side of zero. A decimal point holding the columns in line. An operation's priority. And now an operation's place in a chain. Where it sits is what it means.

That completes our chapter on Foundations of Number. The next chapter is Factors, Multiples and Counting.

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