Predicting how many digits a product will have, before multiplying

0:00Here are three multiplications. Eleven times eleven. One hundred and eleven times one hundred and eleven. And one thousand one hundred and eleven, times itself. Work them out and something lovely happens. One two one. One two three two one. One two three four three two one. The answers climb up and come back down, and each one is a step longer than the last. Here is another. Three times five is fifteen. Thirty three times thirty five is one thousand one hundred and fifty five.

0:39Three hundred and thirty three, times three hundred and thirty five, is one hundred and eleven thousand five hundred and fifty five. Ones, then fives, and one more of each every time. These are genuinely satisfying, and they are the reason this part of the topic exists. Because they are about to teach us something quite different from what they look like. Try this one yourself. One hundred and one, times one hundred and one, is ten thousand two hundred and one.

1:14One hundred and two times itself is ten thousand four hundred and four. One hundred and three times itself is ten thousand six hundred and nine. Look at those three answers and you can almost certainly see the rule. A one. A zero. Then double the small number. Then the small number squared. Two, then four, then six. And one, then four, then nine. Check it on the next one. One hundred and four squared should be one, zero, eight, sixteen. Ten thousand eight hundred and sixteen.

1:50And it is. Four for four. At this point most people would call it a rule. So take one more step. One hundred and five. The rule says: a one, a zero, then ten, then twenty five. Which would be one hundred and one thousand and twenty five. The real answer is eleven thousand and twenty five. Not slightly out. Out by ninety thousand, and a whole digit shorter than predicted.

2:20What broke is that doubling finally needed two columns instead of one, and it carried. Now, four cases fitting was not nothing. It was worth noticing. But it was not evidence that the fifth would fit, and here that is not a technicality. It is simply false. So if examples cannot settle a question about every number, what can? Here is a claim worth settling properly. Take any two two-digit numbers. Any two at all.

2:53Multiply them together. The answer will have either three digits or four digits. Never two. Never five. Now that is a claim about every single pair of two-digit numbers. And your instinct, quite reasonably, is to start trying some. Twelve times thirteen is one hundred and fifty six. Three digits. Eighty seven times ninety four is eight thousand one hundred and seventy eight. Four digits. So far so good. But how long would trying them all take?

3:30There are ninety two-digit numbers. Ten up to ninety nine. So the first number has ninety choices, and so does the second. Ninety times ninety. Eight thousand one hundred pairs. Eight thousand one hundred multiplications to be certain. You will not do that. Nobody will ever do that. And here is the thing that makes this whole topic worth teaching. You do not have to. Two multiplications will settle all eight thousand one hundred of them.

4:03Not sample them. Settle them. Here is the first one. What is the smallest a two-digit number can be? Ten. So the smallest this product could possibly be is ten times ten. Which is one hundred. One hundred has three digits. So no product in the whole collection can be shorter than three digits. That is the floor, and it is now settled for all eight thousand one hundred. Now the ceiling, and this is the clever half.

4:37What is the smallest three-digit number? One hundred. So one hundred times one hundred is ten thousand. And every two-digit number is smaller than one hundred. So every one of our products must be smaller than ten thousand — which means at most four digits. Floor and ceiling. Done. I want to slow down on that second step, because it looks wrong. One hundred is not a two-digit number. It is not one of the cases at all.

5:11So why are we multiplying it? Because a ceiling does not have to be reachable. It only has to be above everything. The largest product we could actually make is ninety nine times ninety nine, which is nine thousand eight hundred and one. That is comfortably under ten thousand, which is exactly what we needed. We could have used ninety nine times ninety nine as the ceiling instead. It is tighter, and it is harder.

5:41One hundred times one hundred you can do in your head, and it is safely above every real case. Choosing a bound that is easy rather than tight is a real skill, and it is most of what makes this argument short. Now the same two multiplications work at any size, so let's ask harder questions. Can a three-digit number times a three-digit number ever give a four-digit answer? Floor first. The smallest three-digit number is one hundred, so the smallest such product is one hundred times one hundred.

6:21Ten thousand. That already has five digits. So no. Not sometimes, not rarely. Never — and we did not test a single case. Try another. Can a four-digit number times a two-digit number give a five-digit answer? Floor: one thousand times ten is ten thousand. Five digits, so yes, it can. Ceiling: nine thousand nine hundred and ninety nine times ninety nine is nine lakh eighty nine thousand nine hundred and one. Six digits.

6:55So that one gives five digits or six. Two answers, both possible, and nothing in between. By now you may have spotted what is going on. Two digits and two digits gave three or four. Three and three gave five or six. Four and two gave five or six as well. In every case, the answer is the two digit counts added together, or one less. And you can see exactly why, from the floor and the ceiling.

7:27The floor multiplies a one followed by some zeros by a one followed by some zeros, and the zeros just add up. The ceiling is a shade under a one followed by all of them. So the product is squeezed between those, and only two lengths fit in the gap. Both really happen, too. Ten times ten gives the short one; ninety nine times ninety nine gives the long one.

7:57So let's use it on cases nobody would attempt by hand. A five-digit number times a five-digit number. Five and five is ten, so the answer has nine digits or ten. An eight-digit number times a three-digit number. Eight and three is eleven. Ten digits or eleven. And one that is frankly absurd. A twelve-digit number times a thirteen-digit one. Twelve and thirteen is twenty five. So the answer has twenty four digits or twenty five.

8:32You have just described the length of a number nobody in this room has ever written down. And you did it with addition. One last thing, and it is the honest part. This rule tells you how long the answer is. It tells you nothing else. It will not tell you the answer. It will not even narrow it much. Of those eight thousand one hundred pairs, there are only just over two and a half thousand different answers.

9:03So knowing a product has four digits leaves thousands of possibilities standing. That is not a weakness. It is what the question asked for. And notice one more thing. Multiplying does not always make a number longer. Two times three is six. One digit times one digit, still one digit. Here is one to try. Pick any two sizes you like, predict the length before you multiply, and then check whether you landed on the short answer or the long one.