Multiplication

Multiplication is a basic number operation that we commonly encounter.

What is multiplication

We mark the operation of multiplication using either a central dot or a cross. If we wanted to write three times seven, we would write it as 3 · 7 or 3 × 7. Sometimes the regular letter x (iks) is used instead of the cross mark: 3 x 7.

We can easily convert multiplication of natural numbers, i.e. the numbers 1, 2, 3, ..., into addition. Consider for example 4 · 6 = ?, four times six. In common parlance we could translate this by saying "go to Franta four times and slap him six times each". What would be the total number of slaps? The first time we slap him six times, the second time we slap him, the third time we slap him, and the fourth time we slap him. The total number of slaps we gave him was 6 + 6 + 6 + 6 = 24. To avoid always having to write out so many of the same number to add, we just introduced multiplication, so if we want to add four sixes, instead of 6 + 6 + 6 + 6 we can just write 4 × 6.

Note that it doesn't matter if we add four sixes or six fours. We always get the same result.

$$ 4 \cdot 6 = 6 + 6 + 6 + 6 = 4 + 4 + 4 + 4 + 4 + 4 = 24 $$

We call the result of multiplication a product, so we say "the product of 4 and 6 is 24", or "the product of 4 times 6 is 24". The numbers we are multiplying, in this case the numbers 4 and 6, are called factors.

Graphical representation of multiplication

Graphically, we can express multiplication as the area of a rectangle. Let's stay with the example 4 · 6. We create a rectangle that has one side of length 4 and another side of length 6. The rectangle will look like this:

Now we count the area of this rectangle, which means we count all the squares that are inside it.

We can see that there are 24 squares. Why? Because we stacked six squares on top of each other four times, so the result is again the sum of 6 + 6 + 6 + 6 = 24.

Multiplying by zero

If we multiply any number by zero, the result will be zero again. The reasoning is simple: if we stick to the previous examples, it's like saying "don't go up to Franta and slap him six times". Franta will be fine this time, because although he is supposed to get six slaps, he is not supposed to get them even once.

It can be shown graphically. A picture showing the product 5 · 0 would look like this:

Since the second factor is zero, one side of the rectangle is also zero, making the rectangle a line segment with no content.

Multiplication priority

Multiplication takes precedence over addition and subtraction. This means that when you have the example 2 + 3 · 4, you calculate the product 3 · 4 = 12 first and then calculate the sum. So after calculating the product, we have 2 + 12 and that equals 14.

If you calculated the sum first, you would get a different result: 2 + 3 = 5 and then 5 · 4 = 20. This result is wrong.

Same with subtraction, so 10 − 8 · 2 + 7 would be counted as if there were parentheses: 10 − (8 · 2) + 7. Thus: 10 − 16 + 7 = 1.

Beware of the tricks that often appear on Facebook. There are examples like 1 + 1 + 1 · 0 = ?. A lot of people see multiplication by zero and immediately write that the result is zero, but that's a bad result. Because multiplication takes precedence, the example is equivalent to 1 + 1 + (1 · 0) = ?.

Here we first multiply the parenthesis: 1 + 1 + 0 = ? and now add it up: 1 + 1 + 0 = 2. The correct result is 2, not 0.

The tricky part is that if you have a simpler calculator, it will also spit out a result of zero. Why? Because in simple calculators you don't enter the whole expression, you enter it one by one and the calculator works with the intermediate result. If you type 1 + 1 + 1 · 0 into such a calculator, the calculator first calculates 1 + 1 = 2, then it calculates 2 + 1 = 3, because it only remembers the intermediate result 2, and finally it calculates 3 · 0 = 0. So the calculator actually calculates the example (((1 + 1) + 1) · 0).

Some calculators can deal with precedence, some allow you to enter the whole example in a piece and then calculate it correctly. If yours can't do that, keep that in mind.

You can read a separate article on multiplication priority.

A little multiplication

For further counting, it is useful to remember the whole small multiplication table, i.e. all multiples of numbers less than 11:

First the small multiplication from one to five:

$$ \begin{array}{lllll} 1 \cdot 1 = 1&2 \cdot 1 = 2&3 \cdot 1 = 3&4 \cdot 1 = 4&5 \cdot 1 = 5 \\ 1 \cdot 2 = 2&2 \cdot 2 = 4&3 \cdot 2 = 6&4 \cdot 2 = 8&5 \cdot 2 = 10 \\ 1 \cdot 3 = 3&2 \cdot 3 = 6&3 \cdot 3 = 9&4 \cdot 3 = 12&5 \cdot 3 = 15 \\ 1 \cdot 4 = 4&2 \cdot 4 = 8&3 \cdot 4 = 12&4 \cdot 4 = 16&5 \cdot 4 = 20 \\ 1 \cdot 5 = 5&2 \cdot 5 = 10&3 \cdot 5 = 15&4 \cdot 5 = 20&5 \cdot 5 = 25 \\ 1 \cdot 6 = 6&2 \cdot 6 = 12&3 \cdot 6 = 18&4 \cdot 6 = 24&5 \cdot 6 = 30 \\ 1 \cdot 7 = 7&2 \cdot 7 = 14&3 \cdot 7 = 21&4 \cdot 7 = 28&5 \cdot 7 = 35 \\ 1 \cdot 8 = 8&2 \cdot 8 = 16&3 \cdot 8 = 24&4 \cdot 8 = 32&5 \cdot 8 = 40 \\ 1 \cdot 9 = 9&2 \cdot 9 = 18&3 \cdot 9 = 27&4 \cdot 9 = 36&5 \cdot 9 = 45 \\ 1 \cdot 10 = 10&2 \cdot 10 = 20&3 \cdot 10 = 30&4 \cdot 10 = 40&5 \cdot 10 = 50 \\ \end{array} $$

And now from six to ten:

$$ \begin{array}{lllll} 6 \cdot 1 = 6&7 \cdot 1 = 7&8 \cdot 1 = 8&9 \cdot 1 = 9&10 \cdot 1 = 10 \\ 6 \cdot 2 = 12&7 \cdot 2 = 14&8 \cdot 2 = 16&9 \cdot 2 = 18&10 \cdot 2 = 20 \\ 6 \cdot 3 = 18&7 \cdot 3 = 21&8 \cdot 3 = 24&9 \cdot 3 = 27&10 \cdot 3 = 30 \\ 6 \cdot 4 = 24&7 \cdot 4 = 28&8 \cdot 4 = 32&9 \cdot 4 = 36&10 \cdot 4 = 40 \\ 6 \cdot 5 = 30&7 \cdot 5 = 35&8 \cdot 5 = 40&9 \cdot 5 = 45&10 \cdot 5 = 50 \\ 6 \cdot 6 = 36&7 \cdot 6 = 42&8 \cdot 6 = 48&9 \cdot 6 = 54&10 \cdot 6 = 60 \\ 6 \cdot 7 = 42&7 \cdot 7 = 49&8 \cdot 7 = 56&9 \cdot 7 = 63&10 \cdot 7 = 70 \\ 6 \cdot 8 = 48&7 \cdot 8 = 56&8 \cdot 8 = 64&9 \cdot 8 = 72&10 \cdot 8 = 80 \\ 6 \cdot 9 = 54&7 \cdot 9 = 63&8 \cdot 9 = 72&9 \cdot 9 = 81&10 \cdot 9 = 90 \\ 6 \cdot 10 = 60&7 \cdot 10 = 70&8 \cdot 10 = 80&9 \cdot 10 = 90&10 \cdot 10 = 100 \\ \end{array} $$