Multiplication is one of the four basic arithmetic operations, alongside addition, subtraction, and division. It involves combining groups of equal size to find the total. It can be represented as a × b, where a is called the multiplicand, b is the multiplier, and the result is called the product.
Multiplication holds many properties some of the important properties of multiplications are:
- Closure Property
- Zero Property
- Commutative Property
- Associative Property
- Identity Property
- Multiplicative Inverse
- Distributive Property
Let's discuss them in detail in context to multiplication.
Closure Property of Multiplication
According to the closure property of multiplication,
“Product of two real numbers is always a real number.”
For example: 4 × 5 = 20, where both 4 and 5 are real numbers, and their product (20) is also a real number.
Zero Property of Multiplication
Zero property of multiplication, also known as the multiplication property of zero, states
"Product of any number and zero is always zero."
Mathematically, the zero property can be expressed as:
a × 0 = 0
for any real number a.
For example: 7 × 0 = 0, -5 × 0 = 0, 0 × 0 = 0, (2/3) × 0 = 0, etc.
Commutative Property of Multiplication
According to the Commutative property of multiplication,
For any two real numbers p and q, their product is always commutative, i.e. a × b = b × c.
For example: 3 × 7 = 21 and 7 × 3 = 21. Here, the order of multiplication does not change the result.
Associative Property of Multiplication
According to Associative property of multiplication,
For three real numbers a, b, and c their product in any order is equal, i.e. (a × b) × c = a × (b × c).
For example: (2 × 3) × 4 = 6 × 4 = 24 and 2 × (3 × 4) = 2 × 12 = 24. The grouping of the numbers does not affect the product.
Identity Property of Multiplication
According to the Identity property of multiplication,
For any real number a, a × 1 = a.
For example: 9 × 1 = 9. Here, multiplying by 1 keeps the original number unchanged.
In the case of dealing with a real number, we have a number which on multiplying with any real number gives the original number itself.
Inverse Property of Multiplication (Multiplicative Inverse)
For real numbers, we have an inverse element of each element such that the product of the two elements is one. Suppose the product of a and b gives the result as one(1), then a and b are called multiplicative inverse of one another.
According to the Inverse Property of Multiplication,
a × (1/a) = 1
For example: 5 × (1/5) = 1. Here, 5 and 1/5 are multiplicative inverses of one another, as their product equals 1.
Note: For any real number, its multiplicative number is always its reciprocal.
Distributive Property of Multiplication
Distributive Property of Multiplication states that when you multiply a number by a sum (or difference), you can distribute the multiplication across each term in the sum (or difference).
- For addition: a × (b + c) = (a × b) + (a × c)
- For subtraction: a × (b − c) = (a × b) − (a × c)
For example: 3 × (4 + 5) = (3 × 4) + (3 × 5) = 12 + 15 = 27
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