Multiplicand: A quantity which is to be multiplied by another (called the multiplier).
Multiplier: A number by which another number (called the multiplicand), is multiplied.
In Mathematics multiplication is written with the symbol \(\times\) (can be called cross sign)
Following are the 7 basic properties to understand the Multiplication:
1. Multiplication as addition: adding the same number again and again:
Count the number of stars:
There are 4 groups and each has 3 stars and 12 stars in total, which means:
4 times 3 gives us 12 as \(\mathbf{4}\times\mathbf{3}=\mathbf{12}\)
Similarly \(\mathbf{2}+\mathbf{2}\\ is the same as\ \mathbf{2}\times\mathbf{2}\):
\(\mathbf{2}+\mathbf{2}=\mathbf{4}\ and\ \mathbf{2}\times\mathbf{2}=\mathbf{4}\)
Also,\(\mathbf{4}+\mathbf{4}+\mathbf{4}\ is the same as\ \mathbf{3}\times\mathbf{4}\):
\(\mathbf{4}+\mathbf{4}+\mathbf{4}=\mathbf{12}\ and\ \mathbf{3}\times\mathbf{4}=\mathbf{12}\)
2. Multiples of Zero (0) and One:
(A) Multiplying by Zero (0): Any number multiplied by 0, gives 0 answer means
\({any}\ {number}\ \times\mathbf{0}=\mathbf{0}\)
Example:
\(1\ \times0=0,\ \ \ \ 15\times0=0,\ \ \ 350\times0=0,\ \ \ 51000\times0=0,\ \ 1\ million\times0=0\ and\ so\ on\)
This rule is called ‘Zero property’
(B) Multiplying by one (1): If we multiply any number with 1, the answer will be the number itself means :
\({any}\ {number}\ \times\mathbf{1}={the}\ {number}\ {itself}\)
Example:
\(1\ \times1=1,\ \ \ \ 15\times1=15,\ \ \ 350\times1=350,\ \ \ 51000\times1=51000,\ \ 1\ million\times1=1\ million\ and\ so\ on\).
This rule is called ‘Identity property’
3. Multiplication Table: Look into the following table and understand the multiplication patterns used:
4. Commutative property of multiplication: Like the addition, Multiplication is also commutative means the order of the factors (‘Multiplicand’ and ‘Multiplier’) doesn’t affect the result.
Two numbers can be multiplied in any order and answer will be the same.
Example
\(Multiplying \mathbf{6}\times\mathbf{2}\ {gives}\ {you}\ {the}\ {same}\ {answer}\ {as}\ {multiplying}\ \mathbf{2}\times\mathbf{6}\)
Another example: Multiplication of 2 with 5 gives \(2\times5=10\)
Now, multiplication of 5 with 2 gives \(5\times2=10\)
The answer in both cases is same, so
\(\mathbf{2}\times\mathbf{5}=\mathbf{5}\times\mathbf{2}=\mathbf{10}\)
Now, look into the above Multiplication table, each answer repeats.
5. Multiplication Tables: Look into the following tables and try to write next few by yourself:
6. Associative Properties: Associative property states that no matter the way you group the factors, the answer (product) will always be the same.
In general write this as:
\(\left(a\times b\right)\times c=a\times\left(b\times c\right);\ where\ a,b\ and\ c\ can\ be\ any\ numbers\)
Example: take three numbers as 3, 5 and 2
Now, \(\left(3\times5\right)\times2=15\times2\)
\(=30\)
And \(3\times\left(5\times2\right)=3\times10\)
\(=30\)
From above two results, we can say that:
\(\left(\mathbf{3}\times\mathbf{5}\right)\times\mathbf{2}=\mathbf{3}\times\left(\mathbf{5}\times\mathbf{2}\right)\)
7. Distributive Properties: The distributive property of multiplication states that when a number is multiplied by the sum of two numbers, the first number can be distributed to both of those numbers and multiplied by each of them separately, then adding the two products together for the same result as multiplying the first number by the sum.
In general, we write this as:
\({a}\times\left({b}+{c}\right)={a}\times{b}+{a}\times{c},\ where\ a,b\ and\ c\ can\ be\ any\ numbers.\)
Example: take three numbers 3, 5 and 2
Now \(3\times\left(5+2\right)=3\times7\)
\(=21\)
And \(3\times5+3\times2=15+6\)
\(=21\)
From the above two results, we can say that
\(\mathbf{3}\times\left(\mathbf{5}+\mathbf{2}\right)=\mathbf{3}\times\mathbf{5}+\mathbf{3}\times\mathbf{2}\)
Distributive property, we use in mental math to find the multiplication.
Example: Use mental math to find the product of \(\mathbf{5}\times\mathbf{23}\)
23 can be written as\(\mathbf{20}+\mathbf{3}\)
So, \(5\times\left(20+3\right)=5\times20+5\times3\)
↓
(multiply 5 with 20 and 3 and then add the products)
\(=100+15=115\)
So, \(\mathbf{5}\times\mathbf{23}=\mathbf{115}\)