A teacher wants to gift $20 cash prize to one of the students for his performance in the class.
In case, if the teacher wants to gift the same cash prize $20 to six students, how much does he have to spend ?
To get answer for the above question, we will be writing 20 six times and add them all.
That is,
20 + 20 + 20 + 20 + 20 + 20
Adding 20 six times can be written as (6x20).
Here, the fact what we have to understand is, the result of adding 20 six times is equal to multiplying 6 and 20.
That is,
20 + 20 + 20 + 20 + 20 + 20 = 6x20 = 120
Therefore, to gift $20 cash prize to six students, the teacher has to spend $120.
Following are the properties of multiplication :
(i) Commutative property
(iii) Associative property
(iv) Identity property of multiplication
(v) Inverse property of multiplication
(vi) Multiplication by zero
(vi) Distributive property of multiplication
Observe the following :
5 x (–6) = –30 and (–6) x 5 = –30
Therefore,
5 x (– 6) = (– 6) x 5
Therefore, multiplication is commutative for all real numbers.
In general, for any two real numbers a and b,
a x b = b x a
Consider the real numbers 2, 5, 7.
Look at
2 x (5 x 7) = 2 x 35 = 70
(2 x 5) x 7 = 10 x 7 = 70
Thus
2 x (5 x 7) = (2 x 5) x 7
So we can say that real numbers are associative under multiplication.
In general, for any real numbers a, b, c,
a x (b x c) = (a x b) x c
Observe the following:
5 x 1 = 5
1 x (- 7) = -7
This shows that ‘1’ is the multiplicative identity for real numbers.
In general, for any real number 'a' we have
a x 1 = 1 x a = a
For any real number, say 'a', the multiplicative inverse is its reciprocal.
That is, multiplicative inverse of 'a' is 1/a.
For example,
The multiplicative inverse of 5 is 1/5
The multiplicative inverse of 3 is 1/3
Note :
Multiplication of a number and its multiplicative inverse is always 1.
That is,
5 x 1/5 = 1
The product of any nonzero real number with zero is zero.
Observe the following:
5 x 0 = 0
–8 x 0 = 0
In general, for any nonzero real number 'a',
a x 0 = 0 x a = 0
Distributive Property of Multiplication Over Addition :
Consider the real numbers 12, 9, 7.
Look at
12 x (9 + 7) = 12 x 16 = 192
12 x (9 + 7) = 12 x 9 + 12 x 7 = 108 + 84 = 192
Thus
12 x (9 + 7) = (12 x 9) + (12 x 7)
In general, for any real numbers a, b, c.
a x (b + c) = (a x b) + (a x c)
Therefore, multiplication is distributive over addition.
Distributive Property of Multiplication Over Subtraction :
Consider the real numbers 12, 9, 7.
Look at
12 x (9 - 7) = 12 x 2 = 24
12 x (9 - 7) = 12 x 9 - 12 x 7 = 108 - 84 = 24
Thus
12 x (9 - 7) = (12 x 9) - (12 x 7)
In general, for any real numbers a, b, c.
a x (b - c) = (a x b) - (a x c)
Therefore, multiplication is distributive over subtraction.
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