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


Real numbers are the numbers we use in everyday life. Whether you're counting your books, measuring the length of a table, or calculating the price of groceries, you're probably using real numbers. Understanding real numbers is fundamental in mathematics and is useful in a variety of practical applications.

What are real numbers?

Real numbers include all the numbers present on the number line. It includes different categories of numbers such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers. These categories together form the group of real numbers.

Real Numbers = Natural Numbers ∪ Whole Numbers ∪ Integers ∪ Rational Numbers ∪ Irrational Numbers

Natural numbers

Natural numbers are the numbers we use for counting. These are the numbers starting at 1 and continuing to infinity: 1, 2, 3, 4, etc. They do not include 0 or any fractions or decimals.

Whole numbers

Whole numbers are like natural numbers, but they include 0. So, whole numbers are 0, 1, 2, 3, 4, etc. Again, they do not include fractions or decimals.

Integers

Integers extend whole numbers to include negative numbers. Thus, integers are ...-3, -2, -1, 0, 1, 2, 3, etc. Integers are considered as the sum of whole numbers and their opposites. Think in terms of form.

-3-2-10123

Rational numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers. A rational number is written in the form a/b, where a and b are integers, and b is not zero. For example:

  • 3/4 (three-fourth)
  • -5/6 (negative five-sixth)
  • 2 (which can be written as 2/1)
  • -10 (which can be written as -10/1)

All integers are rational numbers because their denominator can be 1.

Irrational numbers

Irrational numbers are numbers that cannot be written as simple fractions. They have endless non-repeating decimal parts. Examples include:

  • The square root of 2 (√2), which is approximately 1.414213...
  • The number π (pi), which is approximately 3.141592...

Irrational numbers fill in the "gaps" on the number line that rational numbers do not fill.

012√2

Properties of real numbers

Real numbers obey several fundamental properties that help us perform arithmetic operations and solve mathematical problems. These properties include:

Commutative property

This property states that changing the order of numbers in an addition or multiplication operation does not change the result. For example:

a + b = b + a
a × b = b × a

Example: 3 + 5 = 5 + 3 = 8 or 4 × 7 = 7 × 4 = 28.

Associative property

This property states that when three or more numbers are added or multiplied, the grouping of the numbers does not affect the sum or product. For example:

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

Example: (2 + 3) + 4 = 2 + (3 + 4) = 9 or (5 × 6) × 2 = 5 × (6 × 2) = 60.

Distributive property

This property states that multiplying a number by a sum is the same as doing each multiplication separately. For example:

a × (b + c) = (a × b) + (a × c)

Example: 3 × (4 + 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27.

Identity property

The identity property of addition states that the sum of any number and zero is itself a number, and the identity property of multiplication states that the product of any number and one is itself a number. For example:

a + 0 = a
a × 1 = a

Example: 7 + 0 = 7 or 9 × 1 = 9.

Conclusion

Real numbers are an essential part of mathematics and are highly applicable in real-life situations. They include a wide range of numbers from whole numbers to fractions, as well as non-terminating and non-repeating decimals (irrational numbers). Understanding the properties of real numbers will help you perform arithmetic operations easily and solve various mathematical problems.

Summary example

Let's summarize this with some examples using real numbers:

  • If you add 2.5 (a rational number) and √2 (an irrational number), you still get a real number.
  • An integer such as -7 can be easily added to another integer such as 10 to get 3, which is also a real number.

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Real Numbers

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