Number systems, logic gates, languages notes

A number system is a way of writing numbers with a fixed set of digits and a base (also called radix). The base is how many different digits the system uses. Place value is digit × (base raised to the place). The four systems named here are decimal (base 10), binary (base 2), octal (base 8) and hexadecimal (base 16). A subscript or a prefix marks the base: 36₁₀, 100100₂, 44₈, 24₁₆. Pass mark 36 in decimal is still 36 when the ALU sees 100100 in binary.

Decimal is the human system. Digits 0–9. Places: units, tens, hundreds (10⁰, 10¹, 10²). Anil’s roll 2145 means 2×1000 + 1×100 + 4×10 + 5×1. We use decimal on the OMR sheet and on the mark-list; the CPU does not add in decimal inside the ALU.

Binary is the machine’s system. Digits only 0 and 1, called bits. Places are powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048… A computer is a nest of switches; a switch is on or off, so two symbols are enough. Unit 1’s bit and byte start here: eight bits make one byte. Binary is the same amount written with two digits.

Octal uses digits 0–7. Places are powers of 8: 1, 8, 64, 512… It is a short way to write binary, because 8 = 2³, so three binary bits make one octal digit. Older books and some machine dumps use octal. Digit 8 or 9 cannot appear in a pure octal number.

Hexadecimal (hex) uses sixteen symbols: 0–9 and A–F (A=10, B=11, C=12, D=13, E=14, F=15). Places are powers of 16: 1, 16, 256, 4096… 16 = 2⁴, so four binary bits make one hex digit. Memory addresses and colour codes on a lab PC are often hex because a byte (00 to FF) is two hex digits, not eight binary digits. 36₁₀ = 24₁₆: 2×16 + 4.

Other base → decimal. Expand by place value and add. Binary 100100₂ = 1×32 + 0×16 + 0×8 + 1×4 + 0×2 + 0×1 = 36₁₀. Octal 44₈ = 4×8 + 4×1 = 36₁₀. Hex 24₁₆ = 2×16 + 4×1 = 36₁₀. Hex 2F₁₆ = 2×16 + 15 = 47₁₀.

Decimal → binary, octal or hex. Divide the decimal number by the new base. Write the remainder. Divide the quotient again. Stop when the quotient is 0. Read remainders from bottom to top. Pass mark 36 to binary: 36÷2 → 18 r0; 18÷2 → 9 r0; 9÷2 → 4 r1; 4÷2 → 2 r0; 2÷2 → 1 r0; 1÷2 → 0 r1. Remainders upward: 100100₂. 36 to octal: 36÷8 → 4 r4; 4÷8 → 0 r4 → 44₈. 36 to hex: 36÷16 → 2 r4 → 24₁₆.

Paper maximum 100: 100÷2 chain gives 1100100₂; 100÷8 gives 144₈; 100÷16 gives 64₁₆. Roll 2145: 2145÷2 remainders upward give 100001100001₂ (2¹¹=2048, plus 64, 32 and 1). Same 2145 is 4141₈ and 861₁₆.

Binary ↔ octal. Group binary bits in threes from the right. Pad the left with zeros if needed. Each triad is one octal digit (000=0 … 111=7). 1100100₂ → 001 100 100 → 1 4 4 → 144₈. Reverse: write each octal digit as three bits.

Binary ↔ hex. Group in fours from the right. Each tetrad is one hex digit (0000=0 … 1111=F). 1100100₂ → 0110 0100 → 6 4 → 64₁₆. Reverse: write each hex digit as four bits. A=1010, B=1011, C=1100, D=1101, E=1110, F=1111.

Binary to hex need not pass through decimal: group by four. Octal to hex: octal → binary → hex (or octal → decimal → hex). Example: 44₈ → 100 100₂, pad to 0010 0100 → 24₁₆. Binary never shows the digit 2; octal never shows 8; hex may show A–F.

A logic gate is an electronic circuit that takes one or more binary inputs (0 or 1) and gives a single binary output by a fixed rule. 1 is also called HIGH or TRUE; 0 is LOW or FALSE. The ALU’s decisions are built from gates. Boolean 0/1 is the algebra of these gates; this lesson is the gates themselves, with a truth table for each.

AND. Output is 1 only if every input is 1. Two inputs A, B: 1 only for A=1 and B=1. Classroom: the lab printer prints a mark-sheet only if marks ≥ 36 and fees are paid. One false input, output 0.

OR. Output is 1 if at least one input is 1. 0 only when all inputs are 0. Classroom: the door alarm sounds if the window is open or the door is open.

NOT (inverter). One input. Output is the opposite: 1 becomes 0, 0 becomes 1. A bar over the name (Ā) marks NOT A.

NAND is AND followed by NOT: output is 0 only when every input is 1; otherwise 1. NOR is OR followed by NOT: output is 1 only when every input is 0. NAND and NOR are called universal gates: any other gate can be built from NAND alone, or from NOR alone.

XOR (exclusive OR). Output is 1 if the inputs are different (01 or 10), 0 if they are the same (00 or 11). XNOR (exclusive NOR) is the opposite: 1 if the inputs are the same. XOR is the classroom “odd one”: one switch on, lamp on; both on or both off, lamp off.

A programming language is a coded way to write the instructions the CPU will follow. Unit 1’s generations already named the climb: first generation machine language, second assembly, then high-level. This lesson names the languages and the translators.

Machine language is binary: only 0 and 1, in the exact pattern that one CPU model understands. It needs no translator. It is fast to run and painful to write. A program written for one microprocessor will not run on another without change. First-generation machines were programmed this way.

Assembly language replaces raw bits with short mnemonics: ADD, SUB, MOV, JMP. One assembly line still maps to about one machine instruction. An assembler translates assembly into machine language. The programmer must still know registers and memory. Second-generation work sat here.

High-level language (HLL) looks closer to English and maths: a line can stand for many machine instructions. The same source can be compiled for more than one CPU. FORTRAN, COBOL, BASIC, C, Pascal, Java sit here. A compiler translates the whole program into machine code, then you run the result. An interpreter translates and runs line by line (classic BASIC). A compiler finds many errors before the program runs; an interpreter stops at the first error it meets while running.

Books also call machine language 1GL and assembly 2GL. 3GL (third-generation language) is the classic high-level procedural language: you write how to do the job, step by step — loops, if-then, variables. COBOL, FORTRAN, C, Pascal, Java are 3GL. 4GL (fourth-generation language) sits closer to the human’s what: you name the result more than the steps. Report generators, query languages such as SQL (“list students with marks below 36”), form painters and some database tools are named 4GL in Sinha and Rapidex. 4GL is not “the language of a fourth-generation PC”; a PC can run 3GL. The generation-number of the language is not the generation-number of the hardware.