Multiplication

Elementary multiplication

Multiplication is the third elementary operation of arithmetic discussed in this book. Multiplying natural numbers is basically counting the values of repeated copies of a number together. Counting 3 copies of five equals 15 together: 3 × 5 = 15. In other words: multiplication represents repeated addition:

2 × 5 = 5 + 5

3 × 5 = 5 + 5 + 5

4 × 5 = 5 + 5 + 5 + 5

Verbally 3 × 5 = 15 is expressed as: "three times five equals fifteen" or "five multiplied by three is fifteen". The multiplication sign is generally a cross, "×", but in mathematics or in computer languages also other symbols are often used, such as a (centered) dot (3 · 5 = 15) or an asterisk (3 * 5 = 15). In algebra, the multiplication sign is generally not used at all, when dealing with variables: 3a = 3 × a. Numbers multiplied in a multiplication are called factors and the result is called the product. In 3 × 5 = 15, 3 and 5 are the "factors" and 15 is the "product".

Area of a rectangle 3 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Fig.1. Multiplication 3 × 5 = 15 represented as a stack of 3 rows of 5 squares each, forming a 3 by 5 rectangle with an area of 15 squares.

Figure 1 shows a rectangular area of 15 unit squares as a row of 5 plus a row of 5 plus a row of 5, stacked on each other: 3 × 5 = 5 + 5 + 5 = 15. This same rectangle turned on its other side will show a stack of 5 rows of 3 squares each, or 3 + 3 + 3 + 3 + 3. So apparently: 3 × 5 = 5 × 3. Conclusion: next to addition also multiplication is commutative, meaning that changing the order of the factors does not change the product:

4 × 3 = 3 × 4

3 + 3 + 3 + 3 = 4 + 4 + 4.

Multiplication by zero is always zero, and if the result of a multiplication is zero, then at least one of the factors is zero:

2 × 0 = 0.

0 × 2017 = 0

0 × 0 = 0

If a × b = 0,
then a or b or both must be zero.

Multiplication under ten is typically learned by memorizing the times table or multiplication table (fig.2).

× 1 2 3 4 5 6 7 8 9 10
1 1 2 3 4 5 6 7 8 9 10
2 2 4 6 8 10 12 14 16 18 20
3 3 6 9 12 15 18 21 24 27 30
4 4 8 12 16 20 24 28 32 36 40
5 5 10 15 20 25 30 35 40 45 50
6 6 12 18 24 30 36 42 48 54 60
7 7 14 21 28 35 42 49 56 63 70
8 8 16 24 32 40 48 56 64 72 80
9 9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100
Fig.2. Multiplication table with numbers under ten. The result of an multiplication is at the crossing of a row and column: 2 × 5 = 10.

Multiplying factors > 10

Using the fact that multiplication is commutative (order of factors is arbitrary):

2 × 300 = 2 × 3 × 100 = 6 × 100 = 600.

3 × 40 = 3 × 4 × 10 = 12 × 10 = 120.

30 × 40 = 3 × 4 × 10 × 10 = 12 × 100 = 1200.

10 × 100 (= 10 × 10 × 10) = 1000.

70 × 800 = 7 × 8 × 10 × 100 = 56 × 1000 = 56 000.

30 × 6000 = 3 × 6 × 10 × 1000 = 18 × 10 000 = 180 000.

Distributive property

How to calculate 2 × 13?

With addition we learned that:

700 + 500 = (7+5) × 100 = 12 × 100 = 1200.

Both 7 and 5 are in the ones position, so they can be added, 7 + 5 = 12, and in the hundreds position this makes 1200. We actually applied distribution or the distributive law or the distributive property: (7 + 5) × 100 = 700 + 500. If we apply this to 2 × 13:

2 × 13 = 2 × (10 + 3) = (2 × 10) + (2 × 3) = 20 + 6 = 26.

Some examples:

12 × 10 = (10 + 2) × 10 = 100 + 20 = 120.

3 × 26 = 3 × (20 + 6) = (3 × 20) + (3 × 6) = 60 + 18 = 78.

6 × 78 = 6 × (70 + 8) = (6 × 70) + (6 × 8) = 420 + 48 = 468.

7 × 36 = 7 × (30 + 6) = (7 × 30) + (7 × 6) = 210 + 42 = 252.

4 × 370 = 4 × (300 + 70) = (4 × 300) + (4 × 70) = 1200 + 280 = 1480.

   4 × 370 = 4 × 37 × 10 = ((4 × 30) + (4 × 7)) × 10 = (120 + 28) × 10 = 1480.

3 × 2951 = 3 × (2000 + 900 + 50 + 1) = (3 × 2000) + (3 × 900) + (3 × 50) + (3 × 1) = 6000 + 2700 + 150 + 3 = 8853.

17 × 36 = (10 + 7) × (30 + 6) = ((30 + 6) × 10) + ((30 + 6) × 7) = 360 + (7 × 30) + (7 × 6) = 360 + 210 + 42 = 612.

Multiplication algorithm

The standard multiplication algorithm, also called long multiplication or column multiplication, puts calculation steps applied in the examples above in an orderly, systematic method that always works for any multiplication of two natural numbers:

In the above examples long multiplication is used to calculate 7 × 36 = 252 and 17 × 36 = 612.

The recipe is as follows:
We work from right to left. We start with 7 × 6 = 42. We write the 2 in the ones place of the result row and carry the 4 to the tens column. Then we calculate 7 × 3 = 21 plus the carried 4 makes 25 and we write it in the result row. Done! So, we do not just multiply digits per same column, we multiply the 7 in the lower row by 6 in the ones column, top row and multiply the same 7 by 3 (= 30) in the tens column, top row. Finally we add both results.

To calculate 17 × 36 = 612, we first calculate 7 × 36 as shown, and than, working from right to left, we do the same with the next digit in 17, being 1 (= 10). So, we calculate 1 × 6 and 1 × 3 and write the results in the second result row, but not before we put a 0 most right, because the 1 in the tens position is actually a 10 (10 × 6 = 1 × 6 × 10 and 10 × 30 = 1 × 3 × 10 × 10). Finally we need to add 252 and 360 to find the result of 17 × 36.

This still may look a little complicated, therefore a step by step example:

Multiplication algorithm 1234 × 333 = 410 922. 1 1 1 2 3 4 3 3 3 × 3 7 0 2 1 2 3 4 3 3 3 × 3 7 0 2 3 7 0 2 0 1 2 3 4 3 3 3 × 3 7 0 2 3 7 0 2 0 3 7 0 2 0 0 1 2 3 4 3 3 3 × 3 7 0 2 3 7 0 2 0 3 7 0 2 0 0 + 4 1 0 9 2 2

In this example 1234 × 333 = 410 922 is calculated using long multiplication. The steps in words:

  1. Step 1: The 3 in the ones position times 1234.
    1. 3 × 4 = 12 (carry the 1 to the tens position).
    2. 3 × 3 = 9 plus the carried 1 equals 10 (carry the 1 to the hundreds position).
    3. 3 × 2 = 6 plus the carried 1 equals 7.
    4. 3 × 1 = 3.
  2. Step 2: The 3 in the tens position times 1234.
    1. First write a 0.
    2. Then the same digit calculations as in step 1.
  3. Step 3: The 3 in the hundreds position times 1234.
    1. First write 00 (a zero in the ones, a zero in the tens position).
    2. Then the same digit calculations as in step 1.
  4. Step 4: Add the results of the steps before: 3702 + 37020 + 370200 = 410 922.

Now you try to execute this algorithm with 1234 and 333 swapped, that is, 333 in the top row and 1234 in the row below. You know you must get the same result.

Two more examples:

Suppose you and four of your friends are going to the cinema. One ticket costs $28. How much do the tickets cost together?

Answer

$140,-

Suppose you don't know multiplication yet, could you calculate this by using only the addition algorithm?

Practice application

With the next app you can practice long multiplication. Use the algorithm on a piece of paper to multiply the numbers. Click the exercise or click "CHECK" to see if you did it right.

Exercises:

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