 # Property of Real Number

Updated: 06 Aug 2023

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Real numbers form a fundamental mathematical concept encompassing rational and irrational numbers. Understanding the properties of real numbers is essential for solving equations, simplifying expressions, and mastering various mathematical disciplines. In this article, we will engage in a multiple-choice question (MCQ) format to explore and reinforce our knowledge of the property of real number.

## Property of Real Number

In this article, “property of real number” refer to closure, commutative, associative, identity, and zero properties. These are vital for solving mathematical problems and building a strong foundation in algebra and beyond. In other words, the properties of real numbers are just one of the basic foundations of mathematics.

### Closure Property of Real Number

According to the closure property of Real Number, the sum or product of real numbers is always a real number.

• a+b=Real Number
• a \times b=Real Number

Examples of closure property are

• 3+8=11
• 3 \times 8=24

### Commutative Property of Real Number

According to the commutative property of real number, changing the position or order of two Real Numbers in Adding or Multiplying does not change the final result.

• a+b=b+a
• a \times b=b \times a

Examples:

1. 2+8=8+2
10=10
2. 2 \times 8=8 \times 2
16=16

### Associative Property of Real Number

According to the associative property of real number, changing the grouping of the real numbers in Adding or Multiplying does not change the final result.

• a+(b+c)=(a+b)+c
• a \times (b \times c)=(a \times b) \times c

3+(4+5)=(3+4)+5
3+9=7+5
12=12

Example Multiplication:
3 \times (4 \times 5)=(3 \times 4) \times 5
3 \times 20=12 \times 5
60=60

Zero (0) is called Additive Identity because adding “0” to any real number, the result will be the real number itself.

• a+0=a

Example:
3+0=3

### Multiplicative Identity

1 is called Multiplicative Identity because multiplying “1” to any real number, the result will be the real number itself.

• a×1=a

Example:
3×1=3

If we add a real number to its opposite real number, the result will always be zero (0).

• a+(-a)=0

Example:
2+(-2)=0

### Multiplicative Inverse

If we multiply a real number to its reciprocal real number, the result will always be “1”.

• a \times \frac{1}{a}=1

Example:
-5\times \frac{1}{-5}=1

### Distributive Property of Real Numbers

1. According to the Distributive Property of Real Numbers over Addition states:
• a \times (b+c)=(a \times b)+(a \times c)

Example:
2 \times (4+1)=(2 \times 4)+(2 \times 1)
2 \times 5=8+2
10=10

1. According to the Distributive Property of Real Numbers over Subtraction states:
• a \times (b-c)=(a \times b)-(a \times c)

Example:
2 \times (4-1)=(2 \times 4)-(2 \times 1)
2 \times 3=8-2
6=6

## Properties of Real Numbers MCQs

MCQs provide an engaging and effective way to reinforce our understanding of the properties of real numbers.
Practicing with MCQs can deepen our comprehension of real numbers and enhance our problem-solving skills, making our mathematical journey more enjoyable and rewarding. So, keep exploring and mastering the properties of real numbers to unlock the beauty of mathematics. Let’s embark on this exciting journey of discovery.

1. The set of Real number is the union of two ________ sets.
O Zero
O New
O Disjoint
O None of these

Disjoint
Explanation:

2. Q \cap Q^{\prime}= ________
O Q
O Q^{\prime}
O \emptyset
O All of them

\emptyset
Explanation:
The intersection of Rational and Irrational set is empty set.

3. The sum of two real number is also a real number is called ________ property w.r.t Addition.
O Closure
O Commutative
O Associative
O None of these

Closure
Explanation:
Statement for Closure Property.

4 . The ________ of two real number is also a real number is called closure property.
O Product
O Commutative
O Associative
O None of these

Product
Explanation:
In Closure property the product of two real number is alway be a real number.

5. Example of closure property:
O 7+9=16
O 7 \times 9=63
O Both a & b
O None of these

Both a & b
Explanation:
In Closure property, the sum and product of two real numbers must be the real number. Thus both a & b obey closure property.

6. Commutative property w.r.t addition is ________
O a+b=b+c
O a+c=b+c
O a+b+c=a+b
O a+b=b+c

a+b=b+c
Explanation:
General form of Commutative property w.r.t Addition.

7. Commutative property w.r.t multiplication is ________
O a b=b c
O a c=b c
O a b c=a b
O a b=b c

a b=b c
Explanation:
General form of Commutative property w.r.t Multiplication.

8. Commutative property is ________
O a+b=b+a
O a b=b a
O Both a & b
O None of these

Both a & b
Explanation:
Both a & b Showed the Commutative property of Addition and Multiplication respectively.

9. Associative property w.rt Addition is ________
O a(bc)=(ab)c
O a+(b+c)=(a+b)+c
O Both a & b
O None of these

a+(b+c)=(a+b)+c
Explanation:
General form of Associative property w.r.t Addition.

10. Zero is called ________
O Both a & b
O None of these

Explanation:
Zero (0) is called Additive identity because adding “0” to a number does not change that number.

11. a+0=0+a=a is
O Both a & b
O None of these

Explanation:
Zero (0) is called Additive identity because adding “0” to a number does not change that number.

12. The product of real number and zero is________
O a
O That number
O Imaginary
O Zero

Zero
Explanation:
Any number multiplied to zero is always be zero.

13. 1 is called ________ w.r.t multiplication.
O Multiplicative identity
O Imaginary
O Multiplicative inverse
O None of these

Multiplicative identity
Explanation:
1 is called Multiplicative identity because multiplying “1” to a number does not change that number.

14. a \times 1=1 \times a=a is ________ property.
O Multiplicative identity
O Imaginary
O Multiplicative inverse
O None of these

Multiplicative identity
Explanation:
1 is called Multiplicative identity because multiplying “1” to a number does not change that number.

15. The product of 1 and a number is________
O 10
O Zero
O That number
O None of these

That number
Explanation:
1 is called Multiplicative identity because multiplying “1” to any number does not change that number.

16. The sum of two numbers is zero (0) is called________
O Both a & b
O None of these

Explanation:

17. If a+a^{\prime}=a^{\prime}+a=0 \ then \ a^{\prime} is called ________ of a .
O Both a & b
O None of these

Explanation:
When a real number and its opposite, the result will always be 0.

18. If a+(-a)=-a+a=0 \ then \ -a is called of ________ a .
O Both a & b
O None of these

Explanation:
When a real number and its opposite, the result will always be 0.

19. The product of two numbers is 1 is called________
O Multiplicative identity
O Imaginary
O Multiplicative inverse
O None of these

Multiplicative inverse
Explanation:
When a real number is multiplied by its inverse or reciprocal, the result will always be 1.

20. If a \cdot a^{-1}=a^{-1} \cdot a=1 \ then \ a^{-1} is called ________ of a .
O Multiplicative identity
O Imaginary
O Multiplicative inverse
O None of these

Multiplicative inverse
Explanation:
When the Product of two numbers is “1” then it is said to be Multiplicative inverse.

21 If a \cdot \frac{1}{a}=\frac{1}{a}, a=1 \ then \ \frac{1}{a} is called of ________ a .
O Multiplicative identity
O Imaginary
O Multiplicative inverse
O None of these

Multiplicative inverse
Explanation:
When the Product of two numbers is “1” then it is said to be Multiplicative inverse.

22. Distributive Property of Multiplication over Addition is ________
O a(b+c)=ab+ac
O (b+c)a=ba+ca
O Both a & b
O None of these

Both a & b
Explanation:
Both a & b showed the Distributive Property of Multiplication over Addition

23. If a=a , then it is ________ property.
O Transitive
O Symmetric
O Reflexive
O None of these

Reflexive
Explanation:
Every number is equal to itself is known as Reflexive property.

24. If a=b , then also b=a , it is ________ property.
O Transitive
O Symmetric
O Reflexive
O None of these

Symmetric
Explanation:
By interchanging the sides of an equation doesn’t effect the result is known as symmetric Prperty.

25. If a=b \ and \ b=c then a=c , it is ________ property.
O Transitive
O Symmetric
O Reflexive
O None of these

Transitive
Explanation:
If a equal to b under a rule and b equal to c under the same rules then  a equal to  c is known as transitive property.

26. If y=x^2 \ then \ also \ x^2=y , it is ________ property.
O Transitive
O Symmetric
O Reflexive
O None of these

Symmetric
Explanation:
By interchanging the sides of an equation doesn’t effect the result is known as symmetric Prperty.

27. I x+y=z \ and \ z=a+b then x+y=a+b , it is ________ property.
O Transitive
O Symmetric
O Reflexive
O None of these

Transitive
Explanation:
If a equal to b under a rule and b equal to c under the same rules then  a equal to  c is known as transitive property.

28. If 3=3 , then it is ________ property.
O Transitive>
O Symmetric
O Reflexive
O None of these

Reflexive
Explanation:
Every number is equal to itself is known as Reflexive property.

29. If a=b , then also a+c=b+c , it is ________ property of equality.
O Multiplicative
O Both a & b
O None of these

Explanation:
If we add the same number or expression on both sides of an equation, the equation does not change which means both the sides remain equal.

30. If a=b then also ac=bc , it is ________ property of equality.
O Multiplicative
O Both a & b
O None of these

Multiplicative
Explanation:
If we Multiply the same number or expression on both sides of an equation, the equation does not change which means both the sides remain equal.

31. If a+c=b+c then a=b , it is Cancellation property w.r.t ________
O Multiplication
O Both a & b
O None of these

Explanation:
In this, cancelled the non-zero common factor from both side of the equation by Adding or Subtraction.

32. If ac=bc then a=b , it is Cancellation property w.r.t ________
O Multiplication
O Both a & b
O None of these

Multiplication
Explanation:
In this, cancelled the non-zero common factor from both side of the equation by Multiplication or Divison.

33. Trichotomy property is used for ________ two numbers.
O Increasing
O Decreasing
O Comparing
O Equating

Comparing
Explanation:
See MCQs No. 34

34. Trichotomy property must be true for ________
O a=b
O a > b
O a < b
O All of them

All of them
Explanation:
Trichotomy property is used for compare two numbers.

35. Trichotomy property must be true for________
O 5=5
O 3 < 5
O Both a & b
O None of these

Both a & b
Explanation:
Trichotomy property is used for compare two numbers.

36. If a > b \ and \ b >c \ then \ a > c , it is ________ property of inequality.
O Multiplicative
O Transitive
O All of them

Transitive
Explanation:
If a greater than b under a rule and b greater than c under the same rule then  a greater than  c is known as transitive property of inequlity.

37. If a < b \ and \ b < c \ then \ a < c, it is ________ property.
O Multiplicative
O Transitive
O All of them

Transitive
Explanation:
If a less than b under a rule and b less than c under the same rule then  a less than  c is known as transitive property of inequlity.

38. If a > b then a+c > b+c , it is ________ property of inequlity.
O Multiplicative
O Transitive
O All of them

Explanation:
If we add the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is greater than right side.

39. If a < b then a+c < b+c , it is ________ property.
O Multiplicative
O Transitive
O All of them

Explanation:
If we add the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is less than right side.

40. If x > 5 then ________
O x \times 2 > 5 \times 2
O x \times 2 < 5 \times 2
O Both a & b
O None of these

x \times 2 > 5 \times 2
Explanation:
If we multiply the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is greater than right side.
Note:
The number should be positive.

41. If x > 5 then ________
O x \times -2 > 5 \times -2
O x \times -2 < 5 \times -2
O Both a & b
O None of these

x \times -2 < 5 \times -2
Explanation:
If we multiply the same Negative number to both sides of an inequality, the result will changed. i.e. left side becomes less than right side.

42. For c > 0 \ and \ a < b then Multiplicative property ________
O ac < bc
O ac > bc
O Both a & b
O None of these

ac < bc
Explanation:
If we multiply the same Positive number to both sides of an inequality, the result will remain same. i.e. left side is less than right side.

43. For c < 0 \ and \ a < b then Multiplicative property ________
O ac < bc
O ac > bc
O Both a & b
O None of these

ac > bc
Explanation:
If we multiply the same Negative number to both sides of an inequality, the result will changed. i.e. left side becomes greaer than right side.

44. For c < 0 \ and \ a > b then Multiplicative property ________
O ac < bc
O ac > bc
O Both a & b
O None of these 