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 ↓


Operations on Real Numbers


Real numbers are an important part of mathematics. They include all the numbers we use in our daily lives. These numbers can be thought of as points on an infinite line, commonly known as the number line. The number line extends indefinitely in both directions, with zero in the middle, positive numbers on the right, and negative numbers on the left.



  
  
  0
  
  1
  
  -1

Types of real numbers

Real numbers are divided into various subgroups such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Here is a brief overview of these subgroups:

  • Natural numbers: These numbers are used for counting and start from 1. Example: 1, 2, 3, ...
  • Whole numbers: These include all natural numbers and zero. Example: 0, 1, 2, 3, ...
  • Integers: These are whole numbers and their negatives. Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Rational numbers: Any number that can be expressed as a quotient or fraction p/q where p and q are integers and q ≠ 0. Example: 1/2, 3/4, 5, etc.
  • Irrational numbers: Numbers that cannot be expressed as simple fractions. Their decimal form continues forever without repeating. Example: √2, π, etc.

Basic operations on real numbers

Operations on real numbers are fundamental in mathematics. These operations include addition, subtraction, multiplication, and division. Let us look at these in detail:

Add

Addition is the process of finding the sum of two or more numbers. It is defined by moving to the right on the number line. For example, 5 + 3 = 8 Here's a visual:



    
    
    
    
    +3
    
        
            
        
    
    5
    8

As shown above, starting from 5, moving 3 units to the right on the number line gives 8, which is the result of addition.

Subtraction

Subtraction is the process of finding the difference between two numbers. On the number line, this involves moving to the left. For example, 5 - 3 = 2 looks like this:



    
    
    
    
    -3
    
        
            
        
    
    5
    2

Starting from 5 and moving 3 units to the left gives the result 2, which is the difference.

Multiplication

Multiplication is repeated addition. For example, 3 x 4 means adding 3 four times, which equals 12. Multiplication can be represented visually with jumps on the number line:



    
    
    
    
    
    
    
    
    +3
    
        
            
        
    
    0
    12

Starting from 0, if you add 3 repeatedly, the multiplication result is displayed when the number reaches 12.

Division

Division is the process of dividing a number into equal parts. For example, 12 ÷ 4 = 3 calculates how many times 4 will fit into 12. While division does not have a direct representation on the number line like addition or subtraction, it is understood through division or grouping.

To understand division better, you can look at it as grouping. If you have 12 chocolates and you want to group them into packs of 4, you will have 3 packs. That is exactly what division means.

Properties of real numbers

Operations on real numbers follow specific rules called properties. Understanding these can help solve mathematical problems efficiently.

Commutative property

  • Addition: a + b = b + a. For example, 2 + 3 = 3 + 2.
  • Multiplication: a × b = b × a. For example, 4 × 5 = 5 × 4.

Associative property

  • Addition: (a + b) + c = a + (b + c). For example, (1 + 2) + 3 = 1 + (2 + 3).
  • Multiplication: (a × b) × c = a × (b × c). For example, (2 × 3) × 4 = 2 × (3 × 4).

Distributive property

a × (b + c) = (a × b) + (a × c) This property combines both multiplication and addition. For example, 2 × (3 + 4) = (2 × 3) + (2 × 4) = 6 + 8 = 14.

Identity property

  • Additive identity: The additive identity is 0 because adding 0 does not change the number. So a + 0 = a.
  • Multiplicative identity: The multiplicative identity is 1 because the number does not change when multiplied by 1. So a × 1 = a.

Inverse property

  • Additive inverse: For any real number a, its additive inverse is -a. a + (-a) = 0.
  • Multiplicative inverse: For any non-zero number a, its multiplicative inverse is 1/a. Thus, a × (1/a) = 1.

Ordered operations (BODMAS/BIDMAS)

When performing operations involving more than one number and operation, the rules of "order of operations" help determine what should be done first. This can be remembered by the acronym BODMAS/BIDMAS:

  • B racket
  • O rder or I ndex (exponentiation)
  • D ivision and M ultiplication (left to right)
  • A dd and S ubtract (left to right)

Using BODMAS/BIDMAS ensures that the calculations are done correctly. For example, in the expression 3 + 6 × (5 + 4) ÷ 3 - 7, you would first evaluate what is inside the brackets, then perform multiplication/division and finally addition/subtraction.

Conclusion

Operations on real numbers are fundamental aspects of mathematics that we often encounter in various problem-solving tasks. It is important to understand these operations, their properties, and the order of operations to work effectively with real numbers.


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Decimal Representation of Real Numbers
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Real Numbers on the Number Line

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Operations on Real Numbers

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