Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Number Systems → Real Numbers ↓


Properties of Real Numbers


Real numbers form a broad category in the number system, which includes natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Understanding the properties of real numbers helps to simplify expressions and solve equations efficiently. In this lesson, we will explore various properties of real numbers with lots of examples and explanations.

1. Exchangeable assets

The commutative property states that changing the order of numbers in an addition or multiplication operation does not change the result.

1.1 Commutative property of addition

a + b = b + a

Example:

  • If a = 5 and b = 3, then 5 + 3 = 3 + 5 which both equal 8.
5 + 3 = 3 + 5

1.2 Commutative property of multiplication

a * b = b * a

Example:

  • If a = 4 and b = 6, then 4 * 6 = 6 * 4 which both equal 24.
4 * 6 = 6 * 4

2. Associative property

The associative property shows that the way numbers are grouped in an addition or multiplication operation has no effect on their sum or product.

2.1 Associative property of addition

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

Example:

  • If a = 2, b = 4, and c = 6, then (2 + 4) + 6 = 2 + (4 + 6) both give the result 12.
(2 + 4) + 6 = 2 + (4 + 6)

2.2 Associative property of multiplication

(a * b) * c = a * (b * c)

Example:

  • If a = 3, b = 5, and c = 2, then (3 * 5) * 2 = 3 * (5 * 2) which both calculate to 30.
(3 * 5) * 2 = 3 * (5 * 2)

3. Distributive property

The distributive property connects addition and multiplication. It states that the number multiplied by the sum of two numbers is equal to the sum of the products of that number and each number.

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

Example:

  • If a = 2, b = 3, and c = 4, then 2 * (3 + 4) = 2 * 3 + 2 * 4 both will give result 14.
2 * (3 + 4) = 2 * 3 + 2 * 4

4. Identity property

The identity property of addition states that when you add zero to any real number, it remains unchanged. Similarly, the identity property of multiplication states that any real number remains unchanged when you multiply it by one.

4.1 Identity property of addition

a + 0 = a

Example:

  • If a = 7, then 7 + 0 = 7.
7 + 0 = 7

4.2 Identity property of multiplication

a * 1 = a

Example:

  • If a = 5, then 5 * 1 = 5.
5 * 1 = 5

5. Inverse property

The inverse property states that adding any number to its additive inverse (opposite) gives zero, and multiplying any number by its multiplicative inverse (inverse) gives one.

5.1 Additive inverse

a + (-a) = 0

Example:

  • If a = 9, then 9 + (-9) = 0.
9 + (-9) = 0

5.2 Multiplicative inverses

a * (1/a) = 1

(assume that a ≠ 0)

Example:

  • If a = 8, then 8 * (1/8) = 1.
8 * (1/8) = 1

6. Closing assets

The closure property shows that performing an operation on any two numbers in a set always yields a number within the same set. Real numbers are closed under addition, multiplication, and subtraction but not under division.

6.1 Closure property of addition

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

a ∈ R, b ∈ R ⇒ a + b ∈ R

6.2 Closure property of multiplication

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

a ∈ R, b ∈ R ⇒ a * b ∈ R

7. Additional examples and ideas

Understanding these properties is important for solving algebraic expressions efficiently. Consider another example for the associative property:

Example of associative property

Let us take a = 1, b = 2, and c = 3.

Using the associative property of addition:

  • (1 + 2) + 3 = 3 + 3 = 6
  • 1 + (2 + 3) = 1 + 5 = 6

Example of exchangeable assets

For the commutative property of multiplication:

  • If a = 10 and b = 4, then 10 * 4 and 4 * 10 both equal 40.

Understanding these properties helps to simplify complex equations before performing lengthy calculations. Our goal is to set up formulas and simplify polynomial expressions using these properties.

Conclusion

The properties of real numbers are a foundational aspect of mathematics, helping us to understand and manipulate numbers effectively. From basic operations to solving complex equations, these properties are invaluable tools that simplify calculations and enable a deeper understanding of mathematical concepts. Memorize the key properties: commutative, associative, distributive, identity, inverse, and closure. Mastering these properties can help students confidently tackle various mathematical problems and equations they encounter during their academic journey.


Grade 10 → 1.1.1


U
username
0%
completed in Grade 10

← Prev (1.1)
Real Numbers
Next (1.1.2) →
Rational and Irrational Numbers

Comments 



Properties of Real Numbers

https://www.buddymath.com/g10/1.1.1?bmal=en