Binary number system and bit - byte concepts

3.1.1 · Hardware › Boolean Algebra & Logic Gates

1. Positional number system kya hota hai?

YEH KEY IDEA KYUN HAI: decimal (base-10) bilkul isi tarah kaam karta hai, hum bas itne aadat ho gaye hain ki bhool jaate hain. Binary same rule reuse karta hai with $b = 2$ .

Decimal check (formula KYA kehta hai): $25 3_{10} = 2 \cdot 1 0^{2} + 5 \cdot 1 0^{1} + 3 \cdot 1 0^{0} = 200 + 50 + 3$ Same machinery, $b = 10$ .

2. Binary → Decimal (ek binary number padhna)

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

3. Decimal → Binary (2 se repeated division)

Worked example

$1 3_{10}$ ko binary mein convert karo

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

Remainders bottom-up kyun padhein? Pehla remainder $2^{0}$ bit (rightmost) hai. Aakhri most significant hota hai. Upar se padhne par: $110 1_{2}$ . Verify: $8 + 4 + 0 + 1 = 13$ . ✓

Figure — Binary number system and bit - byte concepts

4. Bit, Nibble, Byte

Data-size prefixes (dhyan rahe):

$1 KiB = 2^{10} = 1024$ bytes (binary prefix).
$1 kB = 1 0^{3} = 1000$ bytes (decimal, SI). Disk makers yeh use karte hain; RAM mein KiB use hoti hai.

6. Forecast-then-Verify drill

Recall Reveal karne se pehle forecast karo

Q1: $2^{10}$ predict karo. → 1024 Q2: Number $200$ store karne ke liye kitne bits chahiye? Forecast: $2^{n} > 200$ chahiye; $2^{7} = 128 < 200$ , $2^{8} = 256 > 200$ → 8 bits. Q3: $10011 0_{2}$ convert karo. Forecast around $32 + 4 + 2 = 38$ ... check: $32 + 0 + 0 + 4 + 2 + 0 = 38$ . Kya aapke forecasts match kiye? Koi bhi gap ab fix karo.

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

101 1_{2}

to decimal.

8 + 2 + 1 = 11

Convert

1101 0_{2}

to decimal.

16 + 8 + 2 = 26

Convert

1 3_{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: ek 12-saal ke bacche ko explain karo

Ek row of light switches imagine karo. Har switch sirf OFF (0) ya ON (1) ho sakta hai. Numbers banane ke liye, hum har switch ko ek "worth" dete hain: sabse rightmost 1 worth hai, agla 2, agla 4, agla 8 — har baar double hota jaata hai. Number padhne ke liye, un switches ki worth add karo jo ON hain. "8" aur "2" aur "1" switches ON karo → $8 + 2 + 1 = 11$ . Ek single switch ek bit hai; eight switches ek line mein ek byte banate hain, aur ek byte koi bhi number 0 se lekar 255 tak dikha sakta hai. Computer aslmein yahi hai: tiny switches ki badi badi rows.

Connections

Boolean Algebra & Logic Gates — gates directly inhi 0/1 bits par operate karte hain.
Hexadecimal number system — binary ko compact writing ke liye nibbles mein group karta hai.
Two's complement — bit patterns use karke negative integers kaise store hote hain.
Logic Gates — AND/OR/NOT binary inputs ko binary outputs mein transform karte hain.
Memory addressing — byte addressable unit kyun hai.

Binary number system and bit - byte concepts

1. Positional number system kya hota hai?

2. Binary → Decimal (ek binary number padhna)

3. Decimal → Binary (2 se repeated division)

4. Bit, Nibble, Byte

5. Steel-manned mistakes

6. Forecast-then-Verify drill

Flashcards

Connections

Concept Map