Property of Real Numbers

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 Numbers

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 Numbers

1. Closure Property of Addition
The sum of real numbers is also a real number.
Statement:
If a, b \in R then a+b \in R
Example:
7+9=16
Where 16 is a real number.

2. Closure Property of Subtraction
The difference between any two real numbers is also a real number.
Statement:
If a, b \in R then a-b \in R
Example:
7-9=-2
Where -2 is a real number.

3. Closure Property of Multiplication
The Product of real numbers is also a real number.
Statement:
If a, b \in R then a . b \in R
Example:
7 \times 9=63
Where 63 is a real number.

4. Closure Property of Division
The quotient of any two real numbers is also a real number but it should be noted that the divisor is not zero.
Statement
For any real numbers a, b \ (where \ b \neq 0 )
If \ a, b \in R \ then \ \frac{a}{b} \in R \ (b \neq 0)
Example:
\frac{7}{9}=0.7777 \ldots
Where 0.7777 \ldots is a real number.

Commutative Property of Real Numbers

1. Commutative Property of Addition:
According to commutative property of Addition states that, if we change the position or order of two Real Numbers in Adding does not change the final result.
Statement:
If a, b \in R then a+b=b+a
Example:
7+9 =9+7
16 =16

2. Commutative Property of Multiplication
According to commutative property of Multiplication states that, if we change the position or order of two Real Numbers in Multiplying does not change the final result.
Statement:
If a, b \in R then a . b=b . a
Example:
7 \times 9 =9 \times 7
63 =63

Note:
Commutative Property w.r.t Subtraction and Division may not be applicable.
Examples:

3. Subtraction
7-9=9-7
-2 \neq 2
Not Applicable

4. Division
\frac{7}{9}=\frac{9}{7}
0.777 \ldots \neq 1.285 \ldots
Not Applicable

Associative Property of Real Numbers

1. Associative Property of Addition:
According to the Associative property of Addition, the order of grouping the real numbers in different ways in Adding does not affect the final result.
Statement:
If a, b, c \in R then a+(b+c)= (a+b)+c
Example:
2+(3+5) =(2+3)+5
2+8 =5+5
10 =10

2. Associative Property of Multiplication
According to the Associative property of Addition, the order of grouping the real numbers in different ways in Multiplying does not affect the final result.
Statement:
If a, b, c \in R then a(b c)=(a b) c
Example:
2(3 \times 5)=(2 \times 3) 5
2(15)=(6) 5
30=30

Note:
Associative Property w.r.t Subtraction and Division may not be applicable.
Examples:

3. Subtraction
2-(3-5)=(2-3)-5
2-(-2)=(-1)-5
2+2=-1-5
4 \neq-6
Not Applicable

4. Division
2 \div(3 \div 5)=(2 \div 3) \div 5
2 \div 0.6=0.66 \div 5
3.333 \ldots \neq 0.132
Not Applicable

Distributive Property of Real Numbers

• According to the Distributive Property of Real Numbers over Addition states:

Statement:
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

• According to the Distributive Property of Real Numbers over Subtraction states:

Statement:
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

Additive Identity of Real Number

Zero (0) is called Additive Identity because adding “0” to any real number, the result will be the real number itself.
Statement:
If a \in R then a+0=0+a=a
Examples:

• 3+0=3
• 3+0=0+3=3
• -5+0=-5
• 9+0=9
• \frac{2}{3}+0=\frac{2}{3}
• 9.5+0=9.5

Multiplicative Identity of Real Number

1 is called Multiplicative Identity because multiplying “1” to any real number, the result will be the real number itself.
Statement:
If a \in R then a \times 1=1 \times a=a
Examples:

• 3×1=3
• 3 \times 1=1 \times 3=3
• -5 \times 1=-5
• 9 \times 1=9
• \frac{2}{3} \times 1=\frac{2}{3}
• 9.5 \times 1=9.5

Additive Inverse of Real Number

If we add a real number to its opposite real number, the result will always be zero (0).
Statement:
If a \in R then a+a^{\prime}=a^{\prime}+a=0 then a^{\prime} is called additive inverse of a
OR
a+(-a)=-a+a=0
10+(-10)=-10+10=0

Example:

• 2+(-2)=0
• 3+(-3)=0
• -5+5=5-5=0
• -20+20=0
• 10-10=0
• -\frac{2}{3}+\frac{2}{3}=0
• \frac{2}{3}+\left(-\frac{2}{3}\right)=0
• \sqrt{2}+(-\sqrt{2})=0
• 9.5-9.5=0

Multiplicative Inverse of Real Number

If we multiply a real number to its reciprocal real number, the result will always be “1”.
Statement:
If a \in R then a . a^{-1}=a^{-1} . a=1 then a^{-1} is called multiplicative inverse of a .
OR
a \cdot \frac{1}{a}=\frac{1}{a} \cdot a=1
OR
a \times \frac{1}{a}=1
10 \cdot \frac{1}{10}=\frac{1}{10} \cdot 10=1
Examples:

• -5\times \frac{1}{-5}=1
• 5 \cdot \frac{1}{5}=1
• -3 \times \frac{1}{-3}=1
• -3\left(\frac{1}{-3}\right)=1
• \frac{1}{3} \times 3=1
• \frac{5}{3} \times \frac{3}{5}=1
• \left(\frac{5}{3}\right)\left(\frac{3}{5}\right)=1
• \left(-\frac{5}{3}\right)\left(-\frac{3}{5}\right)=1
• \left(\frac{-5}{3}\right)\left(\frac{-3}{5}\right)=1
• \sqrt{2}\left(\frac{1}{\sqrt{2}}\right)=1
• 9.5\left(\frac{1}{9.5}\right)=1

Reflexive Property

Every real number or value is equal to itself. e.g. a=a which means that a itself equal to a .
Example

• 5 =5
• \frac{1}{5} =\frac{1}{5}
• -3 =-3
• -3.8 =-3.8
• \sqrt{2} =\sqrt{2}
• 5.9+\sqrt{2} =5.9+\sqrt{2}
• x+y =x+y

Symmetric Property

By interchanging the sides of an equation doesn’t effect the result. e.g. a=b then b=a does not effect the result. In other words, Left side equal to right side of an equation.

• Right side equal to left side of an equation

Example

• 9+7=16 then 16=9+7
• x=16 or 16=x
• x+y=z or z=x+y
• x+2=z or z=x+2
• a-5=b or b=a-5
• 5.9+\sqrt{2}=x or x=5.9+\sqrt{2}

Note:
If x=y then x may be replaced by y or y may be replaced by x in any equation.

Trichotomy Property

By comparing two real numbers, there will be one possibility from the following.

1. a=b
2. a < b
3. a > b

Examples:

1. 6=6
2. 3 < 6
3. 7 > 5

Transitive Property of Real Number

1. Transitive Property of Equality
If a equal to b under a rule and b equal to c under the same rule then a equal to c is known as transitive property.
Statement:
a=b and b=c then a=c
Example:
x+y=z and z=a+b then x+y=a+b
x=5+y and 5+y=z then x=z

2. Transitive Property of Inequality

• 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 inequality.

Statement:
If a > b and b > c then a > c
Example:
If 7 > 5 and 5 > 2 then 7 > \mathbf{2}

• 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 inequality.

Statement:
If a < b and b < c then a < c
Example:
If 3 < 5 and 5 < 7 then 3 < 7

Addition Property of Real Numbers

Addition Property of Equality
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.
Statement:
a=b then a+c=b+c
Examples:
1. x=5 then
x+2=5+2
x+2=7

2. x-3=7
Add 3 on Both sides
x-3+3=7+3
x=10

Addition Property of Inequality
If we add the same number or expression on both sides of an inequality, the inequality does not change.

• If a > b then a+ c > b+c

Examples:
1. 5 > 3
then 5+2 > 3+2
So 7 > 5

2. 5 > 3
then 5+7 > 3+7
So 12 > 10

3. x-3 > 5
Add 3 on B.S
x-3+3 > 5+3
x > 8

• If a < b then a+c < b+c

Examples:
1. 3 < 5
then 3+2 < 5+2
So 5 < 7

2. x-3 < 5
Add 3 on B.S
x-3+3 < 5+3
x < 8

Subtraction Property of Real Number

Subtraction Property of Equality
If we Subtract the same number or expression from both sides of an equation, the equation does not change means both the sides remain equal.
Statement:
a=b then a-c=b-c
Examples:
1. x=5 then x-2=5-2

2. x+3=7
Subtract 3 from Both sides
x+3-3=7-3
x=4

Subtraction Property of Inequality
If we Subtract the same number or expression from both sides of an inequality, the inequality does not change.

• If a > b then a - c > b- c

Example:
1. 5 > 3
then 5-2 > 3-2

2. 5 > 3
then 5-7 > 3-7
So -2 > -4

3. x+3 > 5
Subtract 3 from B.S
x+3-3 > 5-3
x > 2

• If a < b then a-c < b-c

Examples:
1. 3 < 5
then 3-2 < 5-2

2. x < 5
then x-2 < 5-2

3. x+3 < 5
Subtract 3 from B.S
x+3-3 < 5-3
x < 2

4. y+5 < 17
Subtract 5 from Both sides
x+5-5 < 17-5
y < 12

Multiplication Property of Real Numbers

Multiplication Property of Equality
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.
Statement:
a=b then a \times c=b \times c
Examples:
1. x=5
then x \times 2=5 \times 2
So 2x = 10

2. \frac{x}{3}=7
Multiply 3 on Both sides
\frac{x}{3} \times 3=7 \times 3
x=21

Multiplicative Property of Inequality

• If we multiply an inequality by a positive number, then the inequality sign is always remained same.
• When c > 0 :

1. If a < b then a c < b c
2. If a > b then a c > b c

Examples:
1. 5 > 3
then 5 \times 2 > 3 \times 2
So 10 > 6

2. 5 < 7
then 5 \times 2 < 7 \times 2
So 10 > 14

3. \frac{x}{3} > 5
Multiply B.S by 3
\frac{x}{3} \times 3 > 5 \times 3
x > 15
2 x > 24

• If we multiply an inequality by a negative number, then the inequality sign is always changed.
• When c < 0 :

1. If a < b then a c > b c
2. If a > b then a c < b c

Examples:
1. 5 > 3
then 5 \times-2 < 3 \times-2
So -10 < -6

2. 5 < 7
then 5 \times-2 > 7 \times-2
So -10 > -14

3. \frac{x}{-3} < 5
Multiply B.S by -3
\frac{x}{-3} \times-3 > 5 \times-3
x > -15

Division Property of Real Number

Division Property of Equality
If we Divide the same number or expression to both sides of an equation, the equation does not change means both the sides remain equal.
Statement:
If a=b then \frac{a}{c}=\frac{b}{c}
Examples:
1. x=5
then \frac{x}{3}=\frac{5}{3}

2. 2 x=-24
Divide Both sides by 2
\frac{2 x}{2}=\frac{-24}{2}
x=-12

Division Property of Inequality

• If we divide an inequality by a positive number, then the inequality sign is always remained same.
• When c > 0:

1. If \ a < b \ then \ \frac{a}{c} < \frac{b}{c}
2. If \ a > b \ then \ \frac{a}{c} > \frac{b}{c}

Examples:
1. 18 > 12
Then \frac{18}{3} > \frac{12}{3}
So 6 > 4

2. 8 < 12
Then \frac{8}{4} < \frac{12}{4}
So 2 < 3

3. 2 x > 24
Divide B.S by 2
\frac{2 x}{2} > \frac{24}{2}
x > 12

• If we divide an inequality by a negative number, then the inequality sign is always changed.
• When c < 0 :

1. If a > b then \frac{a}{c} < \frac{b}{c}
2. If a < b then \frac{a}{c} > \frac{b}{c}

Examples:
1. 18 > 12
Then \frac{18}{-3} < \frac{12}{-3}
So -6 < -4

2. 8 < 12
Then \frac{8}{-4} > \frac{12}{-4}
So \ -2 > -3

3. -2 x < 24
Divide B.S by -2
\frac{-2 x}{-2} > \frac{24}{-2}
x > -12

Cancellation Property of Real Numbers

Cancellation Property of Addition

1. Cancellation Property of Equality (Addition):
The cancellation property of addition states that if the same real number is added to both sides of an equation, the number can be “cancelled” from both sides of an equation.
Statement:
For any real numbers a, b \& \ c
If a+c=b+c then a=b
Examples:
1. 2 x+5 =9+5
2 x =9

2. 9 x+15 =y+15
9 x =y

2. Cancellation Property of Inequality (Addition)
The cancellation property of addition states that if the same real number is added to both sides of an inequality, the number can be “cancelled” from both sides of an inequality.
Statement
For any real numbers a, b \& \ c

• If a+c > b+c then a > b
• If a+c < b+c then a < b

Examples:
1. x+5 > 9+5
x > 9

2. 9 x+15 < y+15
9 x < y

Cancellation Property of Subtraction

1. Cancellation Property of Equality (Subtraction)
The cancellation property of Subtraction states that if the same real number is subtracted from both sides of an equation, the number can be “cancelled” from both sides of an equation.
Statement:
For any real numbers a, b \& \ c .
If a-c=b-c then a=b
Examples:
1. 2 x-5 =9-5
2 x =9

2. 9 x-15 =y-15
9 x =y

2. Cancellation Property of Inequity (Subtraction)
The cancellation property of Subtraction states that if the same real number is subtracted from both sides of an inequality, the number can be “cancelled” from both sides of an inequality.
Statement
For any real numbers a, b \& \ c

• If a+c > b+c then a > b
• If a+c < b+c then a < b

Examples:
1. x-5 > 9-5
x > 9

2. 9 x-15 < y-15
9 x < y

Cancellation Property of Multiplication

1. Cancellation Property of Equality (Multiplication)
The cancellation property with respect to Multiplication states that if the same non-zero real number is multiplied on both sides of an equation, the number can be “cancelled” from both sides of an equation.
Statement:
For any real numbers a, b \& \ c \ (where \ \ c \neq 0)
If a \times c=b \times c then a=b
OR
If a . c=b . c then a=b
OR
If a c=b c then a=b
Examples:
1. 2 x \times 5=9 \times 5
2 x=9

2. 9 x .15=y .15
9 x=y

3. 3 x=12
3 x=4 \times 3
x=4

4. 4 y=4
y=1

2. Cancellation Property of Inequality (Multiplication)

• The cancellation property of Multiplication states that if the same non-zero Positive real number is multiplied on both sides of an inequality, the number can be “cancelled” from both sides of an inequality and the inequality sign remain unchanged.

Statement
For any real numbers a, b \& \ c \ (where \ \ c \neq 0)

• When c > 0 :

1. If a \times c > b \times c then a > b
2. If a \times c < b \times c then a < b

Examples:
1. x \times 5 > 9 \times 5
x > 9

2. 9 x .15 < y .15
9 x < y

• The cancellation property of Multiplication states that if the same non-zero Negative real number is multiplied on both sides of an inequality, the number can be “cancelled” from both sides of an inequality and the inequality sign is reversed.

Statement:
For any real numbers a, b \& \ c \ (where \ \ c \neq 0)

• When c < 0 :

1. If a \times c > b \times c then a < b
2. If a \times c < b \times c then a > b

Examples:
1. x \times-5 > 9 \times-5
x < 9

2. -9 x < -9 y
x > y

Cancellation Property of Division

1. Cancellation Property of Equality (Division)
The cancellation property of Division states that if the same non-zero real number is divided to both sides of an equation, the number can be “cancelled” from both sides of an equation.
Statement:
For any real numbers a, b \& \ c \ (where \ \ c \neq 0)
\frac{a}{c}=\frac{b}{c} then a=b
OR
If a \div c=b \div c then a=b
Examples:
1. \frac{2 x}{5}=\frac{y}{5}
2 x=y

2. \frac{9 y}{4}=\frac{1}{4}
9 y=1

3. \frac{2 x}{5}=\frac{7}{-5}
\frac{2 x}{5}=-\frac{7}{5}
2 x=-7

2. Cancellation Property of Inequality (Division)

• The cancellation property of Division states that if the same non-zero Positive real number is divided to both sides of an inequality, the number can be “cancelled” from both sides of an inequality and the inequality sign remain unchanged.

Statement
For any real numbers a, b \& \ c \ (where \ \ c \neq 0)

• When c > 0:

1. { If } \ \frac{a}{c} > \frac{b}{c} \ { then } \ a > b
2. { If } \ \frac{a}{c} < \frac{b}{c} \ { then } \ a < b

Examples:
1. \frac{2 x}{5} < \frac{y}{5}
2 x < y

2. \frac{9 y}{4} > \frac{1}{4}
9 y > 1

3. \frac{2 x}{5} > \frac{7}{-5}
\frac{2 x}{5} > -\frac{7}{5}
2 x > -7

• The cancellation property of Division states that if the same non-zero Negative real number is Divided to both sides of an inequality, the number can be “cancelled” from both sides of an inequality and the inequality sign is reversed.

Statement
For any real numbers a, b \& \ c \ (where \ \ c \neq 0)

• When c < 0:

1. If \frac{a}{c} > \frac{b}{c} then a < b
2. If \frac{a}{c} < \frac{b}{c} then a > b

Examples:
1. \frac{8}{-4} > \frac{12}{-4}
8 < 12

2. \frac{18}{-3} < \frac{12}{-3}
18 > 12

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.