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


In mathematics, especially in Class 10 mathematics, it is very important to understand the concept of number system. Number systems form the foundation on which more complex mathematical concepts are built. Numbers are an integral part of our daily lives, whether we realize it or not. They help us count, measure, encode data, solve algebraic equations, and much more. This lesson will tell you about the different types of number systems, their characteristics, examples, visual representations, and applications in simple terms.

What is the number system?

A number system is essentially a writing system for expressing numbers. It is a mathematical notation that represents a given set of numbers in a consistent manner using digits or symbols. There are different types of number systems, and they differ based on the set of digits used, the base of the system, and the rules of arithmetic operations within the system.

The most commonly used number systems are:

  • Natural numbers
  • Whole numbers
  • Integers
  • Rational numbers
  • Irrational numbers
  • Real number
  • Imaginary numbers
  • Complex numbers

Natural numbers

Natural numbers are the numbers we use for counting, like 1, 2, 3, etc. These are also called counting numbers. They start from 1 and go up to infinity.

Natural numbers: 1, 2, 3, 4, 5, ...

Properties of natural numbers:

  • Addition: If you add two natural numbers, the result will be a natural number. For example, 2 + 3 = 5.
  • Multiplication: If you multiply two natural numbers, the result will be a natural number. Example: 2 × 3 = 6.
  • There is no natural number which when multiplied by another natural number gives zero.

Whole numbers

Whole numbers include all natural numbers and zero. So it starts with zero and includes all natural numbers.

Whole numbers: 0, 1, 2, 3, 4, 5, ...

The main difference between natural numbers and whole numbers is the inclusion of zero in the set of whole numbers.

Properties of whole numbers:

  • Closure property: Whole numbers are closed under addition and multiplication. For example, 0 + 3 = 3 and 2 × 4 = 8.
  • Identity: For addition, 0 is the additive identity because 0 + a = a for any whole number a. For multiplication, 1 is the multiplicative identity because 1 × a = a.

Integers

The set of integers includes all whole numbers and their negative counterparts. It includes zero, positive numbers, and negative numbers.

Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...

Properties of integers:

  • Closure property: Integers are closed under addition, subtraction, and multiplication. For example, -2 + 3 = 1 and -4 × 5 = -20.
  • Commutative property: For addition and multiplication, integers are commutative: a + b = b + a and a × b = b × a.
  • Associative property: As for addition and multiplication, integers are associative: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
  • Distributive property: a × (b + c) = a × b + a × c

Rational numbers

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, where q is not zero.

Examples of rational numbers include: 1/2, -3/4, 5, -2 (because 5 = 5/1 and -2 = -2/1).

0 1/2 1

Properties of rational numbers:

  • Closure property: Rational numbers are closed under addition, subtraction, multiplication, and division (except division by zero).
  • Decimal representation: Rational numbers can be either terminating or repeating in their decimal form. For example, 1/4 = 0.25 (terminating) and 1/3 = 0.333... (repeating) are both rational numbers.

Irrational numbers

Irrational numbers are numbers that cannot be expressed as simple fractions, i.e., their decimal expansion does not terminate or repeat. Examples include √2, π and e.

√2 π

Irrational numbers cannot be expressed as fractions p/q with integer values p and q. Their decimal expansion continues forever without repeating.

Examples and properties:

  • The square root of any prime number is irrational (e.g., √3).
  • The constant π, which represents the ratio of a circle's circumference to its diameter, is irrational and is approximately equal to 3.14159...
  • The number e (Euler's number), which is approximately equal to 2.71828..., is also irrational.

Real number

Real numbers include all rational and irrational numbers. This means that they cover every possible value on the number line. Every point on the number line corresponds to a real number.

Examples of real numbers include -2, 0, 1/3, π, √5, 7.9 and -3/2.

Real numbers: -2, 0, 1/3, π, √5, 7.9, ...

Imaginary numbers

Imaginary numbers are numbers that give a negative result when squared. The imaginary unit is represented by i, where i is equal to the square root of -1.

i² = -1

An example of an imaginary number is 4i, 0.5i.

Complex numbers

Complex numbers are numbers that have a real part and an imaginary part. They are expressed in the form a + bi where a and b are real numbers and i is the imaginary unit.

An example of a complex number is 3 + 4i.

a + bi

Conclusion

Number systems are an essential component of mathematics, forming the backbone of many concepts and operations. It is important for students to understand the different types of number systems, as it helps in processing various mathematical problems and real-world calculations.

From ancient calculations to contemporary digital technology, numbers have proven their importance and versatility. Hence, it is essential to study and understand them thoroughly to understand more advanced arithmetic and algebraic concepts.

In school education like Class 10 Maths, a clear understanding of number systems will not only help in exams but also lay a strong foundation for further studies in maths and science. Numbers are the language with which the universe is written, and these systems help us interpret and understand it.


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