Binary number system and bit - byte concepts

1. What is a positional number system?

WHY this is the key idea: decimal (base-10) works this exact way, we're just so used to it we forget. Binary reuses the same rule with $b=2$ .

Decimal check (WHAT the formula says): $253_{10} = 2\cdot10^2 + 5\cdot10^1 + 3\cdot10^0 = 200+50+3$ Same machinery, $b=10$ .

2. Binary → Decimal (reading a binary number)

Set $b=2$ . Column weights are powers of two: $1,2,4,8,16,32,\dots$

3. Decimal → Binary (repeated division by 2)

Figure — Binary number system and bit - byte concepts

4. Bit, Nibble, Byte

Data-size prefixes (be careful):

$1\text{ KiB} = 2^{10} = 1024$ bytes (binary prefix).
$1\text{ kB} = 10^3 = 1000$ bytes (decimal, SI). Disk makers use this; RAM uses KiB.

Q1: Predict $2^{10}$ . → 1024 Q2: How many bits to store the number $200$ ? Forecast: need $2^n>200$ ; $2^7=128<200$ , $2^8=256>200$ → 8 bits. Q3: Convert $100110_2$ . Forecast around $32+4+2=38$ ... check: $32+0+0+4+2+0=\mathbf{38}$ . Did your forecasts match? Fix any gap now.

Flashcards

What base is the binary number system?

Base 2 (only digits 0 and 1).

Why do computers use binary?

Physical switches/transistors have two stable states (ON/OFF), naturally representing 1/0.

In a positional system base

b

, the value of digit

d_i

at position

i

is?

d_i \cdot b^{i}

Convert

1011_2

to decimal.

8+2+1 = 11

Convert

11010_2

to decimal.

16+8+2 = 26

Convert

13_{10}

to binary.

1101 (remainders read bottom-up).

Method to convert decimal to binary?

Repeated division by 2; remainders read bottom-to-top.

What is a bit?

A single binary digit, 0 or 1; smallest unit of data.

How many bits in a byte?

8 bits.

How many bits in a nibble?

4 bits.

How many patterns fit in

n

bits?

2^n

Unsigned range of

n

bits?

0

2^n-1

Max unsigned value in one byte and why?

255, because 256 patterns run 0..255 (0 is counted).

1

KiB equals how many bytes?

2^{10}=1024

bytes.

Why read division remainders bottom-up?

First remainder is the

2^0

(least significant) bit.

Recall Feynman: explain to a 12-year-old

Imagine a row of light switches. Each switch is only OFF (0) or ON (1). To make numbers, we give each switch a "worth": the rightmost is worth 1, next 2, next 4, next 8 — doubling each time. To read a number, add up the worths of the switches that are ON. Turn on the "8" and "2" and "1" switches → $8+2+1=11$ . A single switch is a bit; eight switches in a row make a byte, and one byte can show any number from 0 up to 255. That's all a computer really is: huge rows of tiny switches.

Connections

Boolean Algebra & Logic Gates — gates operate directly on these 0/1 bits.
Hexadecimal number system — groups binary into nibbles for compact writing.
Two's complement — how negative integers are stored using bit patterns.
Logic Gates — AND/OR/NOT transform binary inputs to binary outputs.
Memory addressing — why the byte is the addressable unit.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, computer ke andar bas chhote-chhote switches hote hain jo ya to ON hote hain ya OFF — beech ka koi option nahi. Isiliye computer sirf do symbol samajhta hai: $0$ aur $1$ . Isko hum binary (base-2) kehte hain. Jaise decimal me har column ka value $1, 10, 100$ hota hai, waise hi binary me har column doubling karta hai: $1, 2, 4, 8, 16\dots$ . Number padhne ke liye jahan-jahan $1$ hai, us column ka weight jod do. Example: $1011_2 = 8+0+2+1 = 11$ .

Decimal ko binary banane ke liye repeated division by 2 karo. Har baar remainder ( $0$ ya $1$ ) note karo, aur last me remainders ko neeche se upar padho. Yaad rakhna — top-se-bottom padhoge to number ulta ho jayega, ye sabse common galti hai.

Ek single $0$ / $1$ ko bit bolte hain — sabse chhoti unit. $8$ bits milke ek byte banate hain, aur $4$ bits ek nibble. Important formula: $n$ bits me $2^n$ alag-alag patterns aa sakte hain, isliye unsigned range hoti hai $0$ se $2^n-1$ tak. Isliye ek byte me maximum value $255$ hoti hai, $256$ nahi — kyunki counting $0$ se shuru hoti hai. Ye base samajh liya to logic gates aur poora hardware asaan lagega, kyunki sab kuch inhi $0$ / $1$ pe chalta hai.

step	divide	quotient	remainder (bit)
1	$13/2$	$6$	$1$
2	$6/2$	$3$	$0$
3	$3/2$	$1$	$1$
4	$1/2$	$0$	$1$

Binary number system and bit - byte concepts

1. What is a positional number system?

2. Binary → Decimal (reading a binary number)

3. Decimal → Binary (repeated division by 2)

4. Bit, Nibble, Byte

5. Steel-manned mistakes

6. Forecast-then-Verify drill

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Boolean Algebra & Logic Gates

Connections