Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Overview of Number BASES
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Overview of Number BASES
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Subject: Computer & IT
Class: Senior Secondary 3
Term: 3rd Term
Week: 2
Theme: Coding Systems In Computer
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List digits in the number bases Convert from one number base to another Add and subtract in the number base
A. Definition of Number Base A number base (also known as a radix) defines the number of unique digits, including zero, used to represent numbers in a positional numeral system. The value of each digit depends on its position within the number.
Positional Notation: In any number base, the value of each digit is determined by its face value and its position. Starting from the rightmost digit (least significant digit), each position corresponds to a power of the base. For a number $d_n d_{n-1} ... d_1 d_0 . d_{-1} d_{-2} ...$ in base $b$, its value is: $d_n \times b^n + d_{n-1} \times b^{n-1} + ... + d_1 \times b^1 + d_0 \times b^0 + d_{-1} \times b^{-1} + d_{-2} \times b^{-2} + ...$
B. Common Number Bases Decimal System (Base 10): The system commonly used in everyday life.
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 unique digits).
Example: $(245)_{10} = 2 \times 10^2 + 4 \times 10^1 + 5 \times 10^0 = 200 + 40 + 5 = 245$.
Binary System (Base 2): The fundamental system for computers.
Digits: 0, 1 (2 unique digits). Each digit is called a "bit".
Example: $(1101)_2 = 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 4 + 0 + 1 = (13)_{10}$.
Octal System (Base 8): Used in some computing contexts, especially as a compact representation of binary numbers (3 bits convert to one octal digit).
Digits: 0, 1, 2, 3, 4, 5, 6, 7 (8 unique digits).
Example: $(372)_8 = 3 \times 8^2 + 7 \times 8^1 + 2 \times 8^0 = 3 \times 64 + 7 \times 8 + 2 \times 1 = 192 + 56 + 2 = (250)_{10}$.
Hexadecimal System (Base 16): Widely used in computing for memory addresses, colour codes, and MAC addresses, as it's a very compact way to represent binary (4 bits convert to one hexadecimal digit).
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F (16 unique digits). A = 10, B = 11, C = 12, D = 13, E = 14, F = 15 (in decimal).
Example: $(2AF)_{16} = 2 \times 16^2 + A \times 16^1 + F \times 16^0 = 2 \times 256 + 10 \times 16 + 15 \times 1 = 512 + 160 + 15 = (687)_{10}$.
C. Number Base Conversions Conversion from Any Base to Base 10 (Decimal): Method: Use the positional notation (expanded form) and sum the products.
Rule: Multiply each digit by its corresponding power of the base and sum the results. Worked Example 1.1: Convert $(11011)_2$ to Base 10. $(11011)_2 = 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0$ $= 1 \times 16 + 1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1$ $= 16 + 8 + 0 + 2 + 1 = (27)_{10}$ Worked Example 1.2: Convert $(427)_8$ to Base 10. $(427)_8 = 4 \times 8^2 + 2 \times 8^1 + 7 \times 8^0$ $= 4 \times 64 + 2 \times 8 + 7 \times 1$ $= 256 + 16 + 7 = (279)_{10}$ Worked Example 1.3: Convert $(1F3)_{16}$ to Base 10. $(1F3)_{16} = 1 \times 16^2 + F \times 16^1 + 3 \times 16^0$ Recall F = 15 (decimal). $= 1 \times 256 + 15 \times 16 + 3 \times 1$ $= 256 + 240 + 3 = (499)_{10}$ Conversion from Base 10 (Decimal) to Any Other Base: Method: Repeated division by the target base.
Rule: Divide the decimal number by the target base. Record the remainder. Divide the quotient by the target base again. Continue until the quotient is zero. The new number is formed by reading the remainders from bottom to top (last remainder to first remainder). Worked Example 2.1: Convert $(45)_{10}$ to Binary (Base 2). $45 \div 2 = 22$ remainder $\mathbf{1}$ $22 \div 2 = 11$ remainder $\mathbf{0}$ $11 \div 2 = 5$ remainder $\mathbf{1}$ $5 \div 2 = 2$ remainder $\mathbf{1}$ $2 \div 2 = 1$ remainder $\mathbf{0}$ $1 \div 2 = 0$ remainder $\mathbf{1}$ Reading remainders from bottom up: $(101101)_2$ So, $(45)_{10} = (101101)_2$. Worked Example 2.2: Convert $(178)_{10}$ to Octal (Base 8). $178 \div 8 = 22$ remainder $\mathbf{2}$ $22 \div 8 = 2$ remainder $\mathbf{6}$ $2 \div 8 = 0$ remainder $\mathbf{2}$ Reading remainders from bottom up: $(262)_8$ So, $(178)_{10} = (262)_8$. Worked Example 2.3: Convert $(310)_{10}$ to Hexadecimal (Base 16). $310 \div 16 = 19$ remainder $\mathbf{6}$ $19 \div 16 = 1$ remainder $\mathbf{3}$ $1 \div 16 = 0$ remainder $\mathbf{1}$ Reading remainders from bottom up: $(136)_{16}$ So, $(310)_{10} = (136)_{16}$. Conversion Between Binary, Octal, and Hexadecimal: These conversions are direct because 8 is $2^3$ and 16 is $2^4$. a.
Binary to Octal: Group binary digits into sets of three from right to left, adding leading zeros if necessary. Convert each group of three to its octal equivalent.
Digital Systems and Computing in Nigeria: Number bases are the fundamental language of all digital devices. In Nigeria, this knowledge is critical for understanding how: Mobile Phones and Networks: Every call, SMS, and data packet sent via MTN, Glo, Airtel, or 9mobile is encoded and transmitted using binary. Network engineers and technicians use hexadecimal to identify MAC addresses of devices, troubleshoot network issues, and configure routers.
Banking Systems: Nigerian banks (e.g., GTBank, Zenith Bank) process countless transactions daily using computers. The amounts, account numbers, and transaction details are internally represented in binary. Understanding this helps IT professionals working in financial institutions to secure and manage data effectively.
JAMB CBT Centres: The computer-based testing (CBT) systems used for JAMB examinations operate entirely on binary logic. From storing student biodata to processing exam answers and displaying results, number bases are at the core. A basic understanding helps future computer science students grasp the underlying principles of such systems. Colour Representation in Graphics and Web Design: Nigerian graphic designers or web developers (e.g., creating websites for local businesses, designing event posters) frequently use hexadecimal codes to specify colours (e.g., `#FF0000` for red, `#00FF00` for green). Each pair of hexadecimal digits represents the intensity of red, green, and blue (RGB) components, which are internally processed as binary values. Knowledge of hexadecimal conversions allows for precise colour manipulation and design.
Data Storage and Memory Addressing: When a Nigerian student saves a document, watches a Nollywood movie on their laptop, or copies files to a flash drive, the data is stored in memory locations addressed by binary or hexadecimal values. Debugging software or understanding hardware architecture often requires familiarity with these number systems, which is essential for IT support and hardware engineers in Nigeria.