Contents
In this publication, we will consider 4 basic properties of multiplication of natural numbers, accompanying them with examples for a better understanding of the theoretical material.
Number multiplication properties
Property 1: commutative law
By rearranging the places of the factors, their product does not change.
a ⋅ b = b ⋅ a
examples:
- 5 ⋅ 8 = 8 ⋅ 5
- 14 ⋅ 29 = 29 ⋅ 14
Note: the number of factors can be any. For example, here is the product of three numbers:
Property 2: associative law
The result of multiplying one number by the product of others (for example, the second and third) is equal to the product of the first and second numbers, multiplied by the third.
a ⋅ (b ⋅ c) =
Those. neighboring (and not only) factors (there can be any number of them) can be replaced by their product.
examples:
- 25 ⋅ 4 ⋅ 10 = (25 ⋅ 4) ⋅ 10 =
25 ⋅ (4 ⋅ 10) - 50 ⋅ 2 ⋅ 30 ⋅ 5 =
(50 ⋅ 2) ⋅ (30 ⋅ 5) - 20 ⋅ 6 ⋅ 15 ⋅ 4 ⋅ 11 =
(20 ⋅ 4) ⋅ (6 ⋅ 15) ⋅ 11
Property 3: distributive law
Multiplication by the sum of numbers
To multiply a number by a sum, you need to multiply this number separately by each term, then add the results.
a ⋅ (b + с) = a ⋅ b + a ⋅ c
Factors can be interchanged (according to the commutative property discussed above):
(b + с) ⋅ a = a ⋅ b + a ⋅ c
examples:
- 54 ⋅ (13 + 17) =
54 ⋅ 13 + 54 ⋅ 17 - 16 ⋅ (4 + 22 + 78) =
16 ⋅ 4 + 16 ⋅ 22 + 16 ⋅ 78
Multiplication by the difference of numbers
To multiply a number by the difference, you need to multiply it separately by the minuend and the subtracted, then subtract the second from the first result.
a ⋅ (b – с) = a ⋅ b – a ⋅ c
Swapping the factors, we get:
(b – с) ⋅ a = a ⋅ b – a ⋅ c
examples:
- 9 ⋅ (18 – 5) = 9 ⋅ 18 – 9 ⋅ 5
- (63 – 48 – 20) ⋅ 3 =
63 ⋅ 3 – 48 ⋅ 3 – 20 ⋅ 3
Property 4: multiplication by zero
If a number (product of numbers) is multiplied by zero, the result is zero.
a ⋅ 0 = 0
examples:
- 12 ⋅ 0 = 0
- 24 ⋅ 36 ⋅ 51 ⋅ 0 = 0