Theory
The joke on Chirag's T-shirt
Chirag's Friday T-shirt reads: "There are 10 kinds of people: those who understand binary and those who don't."
The customer laughs politely without getting it. You will, in about four minutes.
Here is the real puzzle behind the joke: the symbols "10" do not always mean ten. What a numeral means depends on the system it is written in, and computers use a different system from you.
Theory
Why circuits count in twos
Look at any switch in the shop: ON or OFF. No half-on. A computer is millions of microscopic switches (transistors), so its natural alphabet has exactly two letters: current flowing = 1, not flowing = 0. Asking a circuit to reliably hold ten voltage levels is asking for errors; two states are crisp and cheap. That is the entire reason binary exists.
Theory
The one rule under every system
All these systems are positional: a digit's value = digit × (base raised to its position), positions counted from 0 at the right.
You already do this in decimal without noticing:
452 = 4·10² + 5·10¹ + 2·10⁰ = 400 + 50 + 2
Same rule, base 2:
101₂ = 1·2² + 0·2¹ + 1·2⁰ = 4 + 0 + 1 = 5
So binary "10" = 1·2 + 0 = two. The T-shirt says "there are TWO kinds of people". Now you are in the first kind.
At a glance
The four systems exams use
| System | Base | Symbols | Where you meet it |
|---|---|---|---|
| Decimal | 10 | 0 to 9 | Everyday life, money |
| Binary | 2 | 0, 1 | All computer internals |
| Octal | 8 | 0 to 7 | Unix file permissions, legacy |
| Hexadecimal | 16 | 0-9 and A-F | Colour codes, MAC addresses |
Theory
Hex: binary's shorthand
Base 16 needs sixteen symbols, so after 9 it borrows letters: A=10, B=11, C=12, D=13, E=14, F=15.
Why bother? Because one hex digit = exactly four binary bits, so long bit-strings compress neatly:
1111 1111₂ = FF₁₆ = 255
You have seen hex all your life: the colour #FF5733 in any design app is three hex pairs (red FF, green 57, blue 33), and your WiFi's MAC address is six hex pairs.
Quiz
Which of these is NOT a valid octal number?
- 382
- 377
- 701
- 125
Show the answer
382
Octal is base 8, so its symbols run 0 to 7 only. The digit 8 in 382 does not exist in octal, like the letter Z in a Hindi word. This "spot the illegal digit" question is an exam favourite; check every digit against the base before anything else.
Think first
Five-second check
In hexadecimal, what is the value of the single digit C? And what decimal value is the hex number 1A? Work both out before revealing.
Show the answer
C = 12 (A=10, B=11, C=12). And 1A₁₆ = 1·16 + 10·1 = 26.
If you said C = 3 (counting C as the third letter), that is the classic slip: the letters START at ten, so count A=10 upward, not A=1.
Watch out
Where marks leak
Reading binary aloud as decimal: 101₂ is five, never "one hundred one". Using digit 8 or 9 inside octal, or forgetting that hex letters start at A=10. And always write the base as a subscript in exams (101₂, 377₈, FF₁₆): without it, the examiner cannot tell which system you mean, and ambiguous answers lose marks.
Theory
Where this goes next
The very next topic converts between decimal and binary both ways, and the one after adds and subtracts in binary, exactly what the ALU in your block diagram does all day. Later, ASCII gives every key on the keyboard a number, and those numbers travel as the bits you just learned to read. Base 2 is the ground floor of the entire course.
Summary
Key takeaways
- Base = how many symbols a system has; positions carry powers of the base.
- Decimal base 10, binary base 2, octal base 8, hexadecimal base 16 (A=10 to F=15).
- Computers use binary because transistors have exactly two reliable states.
- One hex digit packs four binary bits: 1111 1111₂ = FF₁₆ = 255.
- Always mark the base as a subscript in answers.
- Memory hook: digit × base-power, position counts from zero on the right.