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
Multipliers: 10, 1, and 9 (actions: shift and subtract)
Multipliers: 2 and 5 (actions: double and halve)
Multipliers: 3, 4, 6, 7, and 9 (actions: addition and the actions listed in (1) and (2))
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.
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.
Example 10*357
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.
Example 1*357
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.
Example 9*357
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
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.
Example 2*281
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.
Example 5*721
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.
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
Example 3*812
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
Example 4*122
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
Example 6*704
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
Example 7*246
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
Example 8*389
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.
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.