﻿ Multiplication — Math18

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# Multiplication

Content

## What is multiplication?

Multiplication is one of the four basic operations of arithmetic which consists of repeatedly adding a number according to the number of times indicated by another, therefore, it is considered an equivalent operation of the sum since the multiplied number can be expressed equivalently in a sum. For example: 3 x 2 = 6 that corresponds to twice adding the three 3 + 3 = 6 or three times adding the two 2 + 2 + 2 = 6. Multiplication is a thermal with origin in the Latin "multiplicatio" that allows to name the action and the reaction of multiplying.

### Symbol or sign of the multiplication

The representation or sign of the multiplication is known as "multiplied by" and is represented by a cross(x), it can also be represented with a midpoint.

Note: In the absence of these characters, the asterisk (*) is usually used as a sign of multiplication, it is common in computing for programming languages.

## Parts of the multiplication

It is considered 2 important elements but one of these elements contains another 2 elements:

• Coefficient or Factors: Corresponds to the numbers that multiply and this in turn decomposes into two terms:
• Multiplicand: Number that is multiplying or number to be added.
• Multiplier: Times that multiplying must be added.
• Product: It is the result of multiplication.
x
3 ← Multiplicand 2 ← Multiplier / 6 ← Product

Another way to represent the previous multiplication would be: 3 x 2 = 6 (3 is a multiplicand, 2 is a multiplier and 6 is the product).

## Multiplication properties

There are different basic properties that are fulfilled in a multiplication:

• Commutative: “The order of the factors does not alter the product”. For example: 3 x 2 = 6 is equivalent to 2 x 3 = 6.
• Associative: “The order in a multiplication of 3 or more factors does not matter”, For example: 2 x 3 x 2 = 12, I can start by multiplying 2 x 2 = 4 and then 4 x 3 = 12, to check it is possible to multiply 2 x 3 = 6 and then 6 x 2 = 12, this way we see that the same result is obtained.
• Identity or Neutral Element: A number multiplied by 1 will always be the same number, for example: 6 x 1 = 6.
• Distributive: The sum of two numbers multiplied by a third is equal to the sum of each adding by the third number, for example: 2 x (3 + 4) = 2 x 3 + 2 x 4, in both forms you have the same result.
• Multiplicative Property of Zero: Every number multiplied by 0 is always 0, for example 9 x 0 = 0.
• Closing property: The product of two natural numbers results in another natural number, for example 3 x 4 = 12.
• Common factor: It consists of the inverse process of distributive property. If we have several operations either addition or subtraction and have a common factor or equal number, it is possible to transform the sum or subtraction into product by extracting said factor, for example: (3 x 4) + (5 x 4) = 4 x ( 3 + 5), when performing the operation we obtain an equivalent result 12 + 20 = 4 x (8) therefore 32 = 32.

## How do we multiply?

The best way to learn to multiply is by using squares, it is recommended to use a squared notebook to multiply row "↔" by column "↕" so all the boxes contained within correspond to the value of the multiplication.

2 x 3
3 x 2

There are different learning methods for the realization of multiplications, among these methods we can find especially two, which are used for numbers of small quantities and the other method for numbers of large quantities.

• Multiplication arranged in a line: This is used in multiplications of a small amount, as experience is gained increases the ease of this method for larger numbers. It should be considered that for a person who is learning 6 x 8 math it can be a bit confusing, but the purpose is to gradually raise the difficulty. This method helps the learning of mental calculation.

Examples:

2 x 4 = 8
3 x 3 = 9
2 x 3 = 6
6 x 1 = 6

Exercise:

A) 2 x 4 = ?
B) 3 x 2 = ?
C) 4 x 2 = ?
D) 3 x 3 = ?
• Multiply in Column: This is a method for the multiplication of large numbers. The multiplication are placed, one below another in columns which corresponds to the place value of each integer. The place v

Examples:

4 x2 /   8
3 x3 /   9
4 x0 /   0
3 x3 /   9

Exercise:

A)
1 x6 / ?
B)
2 x4 / ?
C)
5 x1 / ?
D)
0 x5 / ?

In some sums we will have the case of a carried number, and this can complicate the operations to be performed, it is recommended to have an order to facilitate multiplication and obtain the correct result. What is a carried number? Assuming we have 13 x 4 = 52, we must first multiply the units 3 x 4 = 12 therefore we have "2" as a unit and "1" as a dozen. The ten obtained “1” would be the “carried” for the next position corresponding to the tens 1 x 4 = 4 and we must add the (carried); therefore, 4 + 1 (carried) = 5 tens, resulting in 2 units and 5 tens corresponding to 52.

Examples:

1 16 x2 / 32
2 15 x4 / 60
12 147   x3 / 441
11 264   x3 / 792

The red number represents the one to be added(carried).

Exercise:

A)
12 x3 / ?
B)
26 x3 / ?
C)
128   x2 / ?
D)
258   x3 / ?

Now that you know how to use the carried we are going to complicate the operations a little by increasing the value of the multiplier to the column of tens.

For the next steps to follow we will use example number 2 that corresponds to 35 x 24 of the "examples" below.

Step 1: The multiplier units are multiplied with the multiplying units. That is, 4 x 5 gives a product of 20, the ones units are placed below and the tens units are placed above as it would correspond to one carried.
Step 2: Multiply the units of the multiplier with the tens of the multiplying. That is, 4 x 3 gives a product of 12 and the tens or taken that resulted from step 1 are added. The result is placed at the bottom (12 + 2 = 14).
Step 3: The tens of the multiplier are multiplied with the units of the multiplier. That is, 2 x 5 gives a product of 10, the units are placed down in the tens column and the tens are placed above as it would correspond to one carried.
Step 4: The tens of the multiplier are multiplied with the tens of the multiplier. That is, 2 x 3 gives a product of 6, it is placed down to the left of the last digit that was placed, if it is carried it must be added (6 + 1 = 7).
Step 5: Once you finish multiplying each digit of the factors, add the resulting amounts according to the resulting arrangement, ones with ones, tens with tens and hundreds with hundreds. In this case the sum of units we have 0, tens 4 + 0 = 4 and hundreds 7 + 1 = 8.

Examples:

+
16 x12 /   32 16 192
+
35 x24 / 140 70 840
+
17 x13 /   41 17 211
+
264   x23 /   792 528 6072

The red number represents the product or result of the multiplication.

Exercise:

A)
12 x23 / ?
B)
26   x43 / ?
C)
128   x52 / ?
D)
258   x123 / ?

## Multiplication with decimals

When multiplying numbers with decimals, the operation is performed in the same way that whole numbers are multiplied, although it is important to know how to place the decimal point in the final product.

For example, to multiply A) 15 x 2.3 and B) 1.5 x 2.3:

To place the point in the final product, the spaces to the right of the decimal point are counted. In the first case there is only one number, this means that the decimal point will be placed a space to the left of the final product.
In the second case there are two numbers to the right of the decimal point, this means that the point will be placed two spaces to the left of the final product.

Examples:

A)
+
2.3   x15 /   115   23   34.5
B)
+
2.3  x1.5 /   115   23   3.45

The red number represents the offset of the point.

Exercise:

A)
8.2    x16 / ?
B)
15.4  x .6 / ?
C)
11.82      x3 / ?
D)
32.86        x5 / ?

## Multiplication table

The following multiplication table is called Pythagorean table. The first row and first column contain the numbers to be multiplied, at the intersection of each row and each column is the product of the column by the row.

X

0

1

2

3

4

5

6

7

8

9

10

0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10
2 0 2 4 6 8 10 12 14 16 18 20
3 0 3 6 9 12 15 18 21 24 27 30
4 0 4 8 12 16 20 24 28 32 36 40
5 0 5 10 15 20 25 30 35 40 45 50
6 0 6 12 18 24 30 36 42 48 54 60
7 0 7 14 21 28 35 42 49 56 63 70
8 0 8 16 24 32 40 48 56 64 72 80
9 0 9 18 27 36 45 54 63 72 81 90
10 0 10 20 30 40 50 60 70 80 90 100

## Important Consideration

It is important to consider that when performing a multiplication operation with additions or subtractions, first the multiplication must be performed unless the addition or subtraction is within a parenthesis, for example: 2 x 4 + 5 first we must do 2 x 4 = 8 and then add 8 + 5 = 13, but if we placed parenthesis to the previous operation 2 x ( 4 + 5 ) first we must do the addition within parenthesis 4 + 5 = 9 and then perform the multiplication 2 x 9 = 18. We must be very careful because the result can be totally different.

Note: If an equation has clustering signs, the braces {} must be removed first, then the square brackets [] and lastly the parentheses ().

## Tricks for multiplications

Here are some techniques that may be useful to facilitate the operations to be performed.

When the multiplication has multiples of 10, 100, 1000, and so on, the different numbers of zero are multiplied and the necessary zeros are added.

Examples:

2 x 10 = 20
3 x 100 = 300
23 x 10 = 230
72 x 100 = 7200

## Checking a multiplication

The multiplication can be checked with its inverse mathematical operation, this is the division. To verify a multiplication, the result is divided by any factor and from this division the other factor must be obtained, indicating that the multiplication was correct.

Examples:

12 x5 / 60
60 /
5
= 12

Taking the result of the multiplication or product (60) and dividing a factor (5) we are going to obtain the result of the other factor (12).

## Multiplication of fractions

For the realization of multiplications of fractions a multiplication of the numerators by the numerators must be done and apart the denominators by the denominators.

Examples:

4 / 3
x
2 / 3
=
4 x 2 / 3 x 3
=
8 / 9