Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Number Systems ↓


Properties of Real Numbers


Real numbers are an essential part of mathematics, forming the foundation for more complex concepts. In Class 9, understanding the properties of real numbers is important as it helps lay the foundation for algebra and beyond. This lesson will provide an in-depth explanation of each property with examples and illustrations to make the concepts easier to understand.

1. Exchangeable assets

The commutative property refers to the order of addition or multiplication not affecting the result. This means that when you change the order of the numbers in an addition or multiplication, the sum or product remains the same.

Add

The commutative property of addition states:

a + b = b + a

3 + 5 = 5 + 3
8 = 8

Multiplication

The commutative property of multiplication says:

a × b = b × a

4 × 7 = 7 × 4
28 = 28
Addition: a + b = b + a Multiplication: a × b = b × a

2. Associative property

The associative property refers to the grouping in addition or multiplication that does not affect the result. This means that when you change the grouping of numbers, the sum or product remains the same.

Add

The associative property of addition states:

(a + b) + c = a + (b + c)

(2 + 3) + 4 = 2 + (3 + 4)
5 + 4 = 2 + 7
9 = 9

Multiplication

The associative property of multiplication states:

(a × b) × c = a × (b × c)

(5 × 6) × 2 = 5 × (6 × 2)
30 × 2 = 5 × 12
60 = 60
(a + b) + c = a + (b + c) (a × b) × c = a × (b × c)

3. Distributive property

The distributive property connects addition and multiplication. It states that multiplying a number by a sum is the same as doing each multiplication separately.

The distributive property states:

a × (b + c) = a × b + a × c

3 × (2 + 4) = 3 × 2 + 3 × 4
3 × 6 = 6 + 12
18 = 18
a × (b + c) = a × b + a × c

4. Identity property

The identity property refers to the fact that addition and multiplication have an identity number that keeps the numbers unchanged when used. For addition, the identity is 0, and for multiplication, it is 1.

Add

The identity property of addition states:

a + 0 = a

7 + 0 = 7

Multiplication

The identity property of multiplication states:

a × 1 = a

9 × 1 = 9
Sum identity: a + 0 = a Multiplication identity: a × 1 = a

5. Inverse property

The inverse property states that every number has an opposite (additive inverse) or inverse (multiplicative inverse) that brings the result to the identity element.

Add

The inverse property of addition states:

a + (-a) = 0

4 + (-4) = 0

Multiplication

The inverse property of multiplication says:

a × (1/a) = 1 (where a ≠ 0)

5 × (1/5) = 1
Additive inverse: a + (-a) = 0 Multiplicative inverse: a × (1/a) = 1

6. Closing assets

The closure property states that for a group of numbers, the operation of addition or multiplication will always give a result that belongs to the same group.

For example, the set of real numbers is closed under addition and multiplication.

Add

If a and b are real numbers, then a + b is also a real number.

-3 + 2 = -1

Multiplication

If a and b are real numbers, then a × b is also a real number.

-4 × 5 = -20
Completion in addition and multiplication

Conclusion

Understanding these properties of real numbers is important for solving algebraic expressions and equations. They help simplify expressions and provide insight into the behavior of numbers when undergoing various operations. Mastering these properties will lay a strong mathematical foundation for advanced topics in mathematics. Always remember to recognize and use these properties when performing arithmetic operations to make problem-solving easier.


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Properties of Real Numbers

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