Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 ↓


Number Systems


Introduction

In mathematics, we use different ways to represent numbers and perform operations on them. These different ways of representing numbers are collectively known as number systems. Understanding number systems helps us learn how to work with numbers effectively. In this guide, we will explore the basic concepts, the different types of number systems, and how to convert between them.

What is the number system?

A number system is a way of expressing numbers using a consistent set of symbols and rules. The number system helps in performing mathematical operations such as addition, subtraction, multiplication and division. Different cultures have developed different number systems over time, but the most common and widely used is the decimal system.

Types of number systems

There are many types of number systems, but the four most common are:

  1. Decimal Number System (Base 10)
  2. Binary Number System (Base 2)
  3. Octal number system (base 8)
  4. Hexadecimal number system (base 16)

Decimal Number System (Base 10)

The decimal number system is the one we use in our daily lives. It is based on ten different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Each digit has a different place value depending on its place in the number. For example, the number 245 can be expanded as follows:

245 = 2 * 100 + 4 * 10 + 5 * 1
Hundreds place: 2 Tens place: 4 Unit Location: 5

Binary Number System (Base 2)

The binary number system is mainly used in computers and digital systems. It has only two digits, 0 and 1. Each digit in a binary number is called a 'bit'. Binary numbers are important because computers work using the binary system. An example of a binary number is as follows:

The binary number 1011 can be expanded as follows:

1011 (Binary) = 1*2^3 + 0*2^2 + 1*2^1 + 1*2^0 = 8 + 0 + 2 + 1 = 11 (Decimal)
8 0 2 1

Octal number system (base 8)

The octal number system uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. It is sometimes used in computing. An example of an octal number is:

345 (octal) = 3*8^2 + 4*8^1 + 5*8^0 = 192 + 32 + 5 = 229 (decimal)
192 32 5

Hexadecimal number system (base 16)

The hexadecimal number system is used in computing and digital electronics. It uses sixteen unique digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. The letters A through F represent the numbers ten through fifteen. Here's an example of a hexadecimal number:

2F (hexadecimal) = 2*16^1 + 15*16^0 = 32 + 15 = 47 (decimal)
32 15

Conversion between number systems

Decimal to binary

To convert a decimal to binary, divide the decimal number by 2 and record the remainder. Keep dividing the quotient by 2 until you reach zero. The binary number is the remainder read from bottom to top. For example, converting 13 to binary:

13 / 2 = 6 remainder 1
6 / 2 = 3 remainder 0
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1

Binary: 1101

Binary to decimal

To convert from binary to decimal, multiply each bit by 2, which is the power of its position, counting from right to left, starting at 0. Then sum all the values. For example, converting 1101 to decimal:

1*2^3 + 1*2^2 + 0*2^1 + 1*2^0 = 8 + 4 + 0 + 1 = 13

Decimal to octal

To convert a decimal to octal, divide the decimal number by 8 and record the remainder. Keep dividing the quotient by 8 until you reach zero. The octal number is the remainder read from bottom to top. For example, to convert 65 to octal:

65 / 8 = 8 remainder 1
8 / 8 = 1 remainder 0
1 / 8 = 0 remainder 1

Octal: 101

Octal to decimal

To convert from octal to decimal, multiply each digit by 8, which is the power of its position, counting from right to left, starting at 0. Then sum all the values. For example, to convert 101 to decimal:

1*8^2 + 0*8^1 + 1*8^0 = 64 + 0 + 1 = 65

Decimal to hexadecimal

To convert decimal to hexadecimal, divide the decimal number by 16 and record the remainder. Keep dividing the quotient by 16 until you reach zero. A hexadecimal number is the remainder read from bottom to top. For example, to convert 255 to hexadecimal:

255 / 16 = 15 remainder 15 (f)
15 / 16 = 0 remainder 15 (F)

Hexadecimal: FF

Hexadecimal to decimal

To convert hexadecimal to decimal, multiply each digit by its place in the power of 16, counting from right to left, starting at 0. Letter values are used for the digits A through F. For example, converting FF to decimal:

15*16^1 + 15*16^0 = 240 + 15 = 255

Conclusion

Understanding number systems is important in a variety of fields, especially in computing and digital electronics. Different bases such as binary, octal, and hexadecimal provide simpler ways to handle data on a computer, while the decimal system remains central in everyday arithmetic. Becoming comfortable with conversions between these systems enhances analytical skills and deepens your understanding of the structural similarities and differences between them.


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