Short Multiplication


Multiplication with multipliers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and the number 10, has traditionally been called "short multiplication". It puts no limit on the length of the multiplicands. The most common way to teach it during the nineteenth century and the first half of the twentieth century required that students could recall multiplication "facts" up to 9 times 9 instantly, and that they were able, for fairly long multiplicands, to "carry tens" mentally without jotting them down.

The "standard" written algorithm that required this skill is fast, concise, and efficient. But it required years of practice to master (from 4th to 8th grade), and it was used mainly by people who needed to keep business books, namely, shopkeepers and accountants.

Here we present another method, which handles each multiplier differently, and is more complex, but it doesn't require instant recall of "facts", and limits "carries" to cases that are simple and easy to learn.

The Order in Which to Learn Short Multiplication

  1. Multipliers: 10, 1, and 9 (actions: shift and subtract)
  2. Multipliers: 2 and 5 (actions: double and halve)
  3. Multipliers: 3, 4, 6, 7, and 9 (actions: addition and the actions listed in (1) and (2))
  4. Multiplier: 0

In all examples below we show how to carry out the computation on a counting board. But all these methods can also be adapted to written and mental calculations.

Notation: Here, M stands for a multiplier represented in base 10 on a counting board, or verbally, or in writing. In the following examples, note that Board M is always shown on top and board P is always on the bottom.

  1. Three Cases: 10*M, 1*M, and 9*M

    Multiplying by 10 does not depend on any mathematical properties, but on the fact that we write numbers in base 10. So it is achieved by shifting all digits one place to the left and appending the digit 0 at the end.


    Board M holds 357. Because 10 is the multiplicand, Board M is shifted one place to the left. Due to this shift, Board P holds 3570.

    1*M = M is a part of the mathematical definition of 1; we simply copy M into P.


    Board M holds 357. Board P holds a copy of Board M, which is 357.

    Finally, 9 = 10 - 1, so 9*M = 10*M - M.


    Board M holds 357. In the left image, Board P holds 10*357 using white tokens and -1*357 using red tokens. In the right image, regrouping is performed and Board P holds 3213.

    This method is also very useful in mental arithmetic. For example,

               Think: 9*17 → 170 - 17 → 160 - 7 → 153

  2. The Two Cases 2*M and 5*M

    2*M is doubling. It has often been treated as a separate operation, and not as a special case of addition, M + M = 2*M, since it is simpler because "you never need to carry over".   5*M = 10*M/2. But in base 10, 10*M is computed by shifting digits, so multiplication by 5 reduces to halving. And halving again has been treated as an operation different from division, because "you never need to borrow over".

    To double a number in base ten, you scan it from left to right, double each digit, take its unit part, look at the next digit to the right, and if it is 5 or more, you add 1 to the unit part. It is helpful if you memorize the doubles of all digits. But when you are computing on a counting board, you may just double the tokens that represent a digit and regroup.


    Board M holds 281. In the left image, Board P holds 281 doubled using tokens. In the right image, regrouping is performed and Board P holds 562.

    To halve an even number in base ten, you again scan it from left to right, you take the whole part of half as the result and add 5 to it if the previous digit was odd.


    Board M holds 721 and is shifted one place to the left because 10 is the multiplicand. Board P holds 7210.

    The current number held on Board P, 7210, is halved and Board P now holds 3605.

    This method requires that the student can mentally compute half of each whole number from 0 to 9. But this is not hard.

  3. The Cases 3*M, 4*M, 5*M, 6*M, 7*M, and 8*M

    In all cases, you combine the actions described above. P is a partial sum. A minus sign in front of the multiplier means that the number on the counting board consists of red counters.

    3*M: 2*M → P; P+M → P


    Board M holds 812. Board P holds 2*812 = 1624.

    Board M is added to the current number held on Board P so that Board P holds 2436.

    4*M: 2*M → P; 2*P → P       or       5*M → P; P + -M → P


    Board M holds 122. Board P holds 2*122 = 244.

    The current number held on Board P is put on board M. This number is doubled and Board P holds 488.

    6*M: 5*M → P; P + M → P


    Board M holds 704. Board P holds 5*704 = 3520.

    Board M, 704, is added to the current number held on Board P so that Board P holds 4224.

    7*M: -2*M → P; P + -M → P; P + 10*M → P


    Board M holds 246. Board P holds -2*246 = -492 using red tokens.

    Board M, 246, is added opposite to the current number held on Board P using red tokens. Board P now holds -738.

    10*Board M, or 10*246 = 2460, is added to Board P so that Board P holds 1722.

    8*M: -2*M → P; P + 10*M → P


    Board M holds 389. Board P holds -2*389 = -778 using red tokens.

    10*Board M, or 10*389 = 3890, is added to Board P so that Board P holds 3112.

  4. The Case 0*M

    Because 0*M = 0, and an empty counting board represents 0, no action is taken and P = 0.

    This is the easiest case but operations involving 0 are often mysterious. Zero joined the set of counting numbers only in modern times, and counting numbers were then renamed "whole numbers". But even now, division by 0, and 0^0 remain undefined, because any attempt at a definition seems to contradict other accepted properties of numbers.

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