# What are the different properties of numbers?

The arithmetic value which is used for representing the quantity and used in making calculations are defined as Numbers. A symbol like 4, 5, 6 which represents a number is known as a numeral. Without numbers, counting things is not possible, date, time, money, etc, are not possible to count, these numbers are also used for measurement and used for labeling.

### Number System

A number system** **is a method of showing numbers by writing, which is a mathematical way of representing the numbers of a given set, by using the numbers or symbols in a mathematical manner. The writing system for denoting numbers using digits or symbols in a logical manner is defined as a number system. The numeral system is something that Represents a useful set of numbers also reflects the arithmetic and algebraic structure of a number and Provides standard representation.

**Note ** With the help of digits from 0 to 9, all the numbers can be formed. With these digits, anyone can create infinite numbers. For example, 156, 3907, 3456, 1298, 784859 etc

### Properties of Numbers

The properties of numbers make them helpful in performing arithmetic operations on them. These numbers can be written in numeric forms and also in words**. **For example, 3 is written as three in words, 35 is written as thirty-five in words, etc. Students can write the numbers from 1 to 100 in words to learn more. There are different types of numbers, which can be learned. They are whole and natural numbers, odd and even numbers, rational and irrational numbers, etc

These properties of numbers are stated for** real numbers,** the combinations of both the rational and irrational numbers can be defined as real numbers. It can be positive or negative and is denoted by the symbol “R”. The different properties of numbers are,

**Commutative Property**

It is applicable only for addition and multiplication processes. it means at the time of adding or multiplying any two numbers can swap their positions but will give the same result. If a and b are two real numbers, then according to commutative property,

Commutative property of addition: In this property, when two integers are added, the answer will remain unchanged even if the position of the numbers is changed.

Commutative property of multiplication: In this property, when multiplying two real numbers, the answer after multiplication will remain the same, even if the position of the integers is interchanged.

**a+b = b+a {Commutative property of Addition}**

**a.b = b.a {Commutative property of Multiplication}**

**Associative Property **

As per the Associative Property, does not matter how the numbers are grouped, they can be added or multiplied together and will get the same result or answer. In other words, the placement does not matter when it comes to multiplying or adding. Associative property can only be used with addition and multiplication but we cannot use it with subtraction or division. If a, b and c are three real numbers, then as per the associative property,

**(a + b) + c = a + (b + c) {Associative property of addition}**

**(a × b) × c = a × (b × c) {Associative property of multiplication}**

**Distributive Property **

As per the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together. If p, q, and m are three real numbers, then as per the distributive property,

p × (q + m) = (p× q) + (p × m)

**Example:** 2 × (3 + 4) = 2×3 + 2×4

2 × 7 = 6 + 8

14 = 14

The distributive property helps in making problems simpler and can be used as the distributive property of multiplication to rewrite expression by distributing or breaking down a factor as a sum or difference of two numbers. Here, for example, calculating 8 × 20 can be made easier by breaking down 20 as 13 + 7 or 30 − 10

**The distributive property of multiplication over addition: **

8 × (13 + 7)

= 8 × 13 + 8 × 7

= 104 + 56

= 160

**The distributive property of multiplication over subtraction:**

8 × (30 − 10)

= 8 × 30 − 8 × 10

= 240 − 80

= 160

The outcome from both the methods will be the same i.e, 160

**Closure Property**

If a number is added to another number, then the outcome will be a number only, such as p + q = m; where p, q, and m are three real numbers

Example, 2 + 2 = 4

**Identity Property **

If zero is added to a number then Zero is the additive identity, by this, the number will remain unchanged. The real number which is multiplied to one (1) is equal to the number itself. The number one is the multiplicative identity,

**a +0 = a {additive identity}**

**a × 1 = a {multiplicative identity}**

Example: 6 + 0 = 6 and 6 x 1 = 6

**Additive Inverse**

An additive inverse of a number is defined as the value while adding with the original number results in zero value. It’s the value, adding to a number to yield zero. Here **a** is the original number, the **additive inverse of a** will be minus a i.e,**-a,** such that,

**a + (-a) = a – a = 0**

**If an original number is added to its own negative number, then the result will be zero.**

**a + (-a) = 0**

Example: 4 + (-4) = 4 – 4 = 0

**Multiplicative Inverse **

The multiplicative inverse of a number for any n is simply 1/n. It is represented as: 1 / x and also called as the reciprocal of a number and 1 is called the multiplicative identity. Multiplicative inverse natural numbers can be found easily, but it’s difficult for complex and real numbers.

**For example**, the multiplicative inverse of 4 is 1/4, 46 is 1/46, 134 is 1/134, 8 is 1/8, etc, whereas the reciprocal of 0 will give an infinite value or 1/0 = ∞. Now to check the inverse of a number is correct or not, we can perform some equations such that,

- 3 × 1/3 = 1
- 47 × 1/47 = 1
- 13 × 1/13 = 1
- 8 × 1/8 = 1

**Zero Product Property **

Zero product property, also called as zero product principle states that if f × g = 0 , then f = 0 or g = 0 or both f = 0 and g = 0 .

Example: 8 x 0 = 0 or 0 x 8 = 0

### Sample Problems

**Question 1: Give examples for Commutative properties of numbers?**

**Answer:**

Commutative property of addition

- 4 + 3 = 3 + 4 = 7
- 15 + 11 = 11 + 15 = 26
Commutative property of multiplication

- 8 × 7 = 7 × 8 = 56
- 15 × 3 = 3 × 15 = 45

**Question 2: Which property is this equation: (3+ 4) + 2 = 3 + (4 + 2)?**

**Answer:**

The equation shown is the Associative property of addition. The value of the equation is 9.

**Question 3: Discuss with an example how distributive property makes calculations easier.**

**Answer:**

The distributive property does help in making the calculations easier. For instance, the calculation is 5 × 25. It can be broken to form 5 × (20 + 5), solving this would be easier using distributive property,

5 × (20 +5)

= (5 × 20) + (5 × 5)

= 100 + 25

= 125

**Question 4: What is the multiplicative inverse for,**

**3****1/9****22**

**Answer:**

The multiplicative inverses are,

- The multiplicative inverse of 3 is 1/3.
- The multiplicative inverse of 1/9 is 9.
- The multiplicative inverse of 22 is 1/22.

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