Binary number system — conversions from - to decimal, counting in binary

1.1.1 · Coding › How Computers Work

WHY binary at all?

WHAT is place-value (har base ke peeche ka engine)

Scratch se derivation (pehle decimal, taaki idea familiar lage): "253" do-sau-tirpan kyun hai? Kyunki humne agree kiya hai ki har slot apne right wale slot se das guna zyada worth hai: $253 = 2 \cdot 1 0^{2} + 5 \cdot 1 0^{1} + 3 \cdot 1 0^{0} = 200 + 50 + 3$ Ye step kyun? Sabse baayi digit "units se" do positions door hai, isliye wo $1 0^{2}$ se scale hoti hai.

Ab base 10 → base 2 karo. Sirf itna badlega ki weights powers of 2 ban jaate hain, aur allowed digits sirf ${0, 1}$ reh jaate hain:

HOW to convert Binary → Decimal

Worked example

$101 1_{2}$ ko decimal mein convert karo Bits ke upar weights likho:

bit	1	0	1	1
weight	$2^{3} = 8$	$2^{2} = 4$	$2^{1} = 2$	$2^{0} = 1$

Jahan bit 1 hai, unke weights add karo: $8 + 0 + 2 + 1 = 11$ Ye step kyun? 0 kuch contribute nahi karta; 1 apna place weight contribute karta hai. Toh aap bas "on" columns ko sum karte ho. Answer: $101 1_{2} = 1 1_{10}$ .

HOW to convert Decimal → Binary (repeated division)

Worked example

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

$N$	$N \div 2$	quotient	remainder
13	6 r 1	6	1
6	3 r 0	3	0
3	1 r 1	1	1
1	0 r 1	0	1

Remainders bottom→top padhna: $1101$ . Bottom-to-top kyun? Pehla remainder units bit hai ( $2^{0}$ ), aakhri highest bit hai. Upar ki taraf padhne par highest weight pehle aata hai, jaise hum numbers likhte hain. Check: $8 + 4 + 0 + 1 = 13$ ✓

Counting in binary

Worked example 3 bits mein 0 se 7 tak count karo

Ye step kyun? Dekho rightmost bit har step mein flip karti hai (1,0,1,0…), middle bit har 2 steps mein, left bit har 4 steps mein — har column apne right wale se aadha baar flip karta hai, kyunki uska weight double hai.

Figure — Binary number system — conversions from - to decimal, counting in binary

Forecast-then-Verify

Recall Forecast:

$10 0_{10}$ likhne ke liye kitne bits chahiye? Phir verify karo. Forecast: $2^{6} = 64$ , $2^{7} = 128$ . Kyunki $64 \leq 100 < 128$ , humein 7 bits chahiye. Verify (repeated division): $100 \to 50$ r0, $50 \to 25$ r0, $25 \to 12$ r1, $12 \to 6$ r0, $6 \to 3$ r0, $3 \to 1$ r1, $1 \to 0$ r1 → bottom-up $1100100$ . Ye 7 digits hai ✓ aur $64 + 32 + 4 = 100$ ✓.

Common mistakes

Flashcards

Binary system kaunsa base use karta hai, aur ye computers ke liye natural kyun hai?

Base 2; computer switches ke paas exactly do stable states hote hain (off=0, on=1), jo saste aur noise-resistant hain.

Binary→decimal: position

i

(rightmost = 0) par bit ka weight kya hota hai?

2^{i}

101 1_{2}

ko decimal mein convert karo.

8 + 0 + 2 + 1 = 11

11010 0_{2}

ko decimal mein convert karo.

32 + 16 + 4 = 52

Decimal ko binary mein convert karne ka algorithm kya hai?

2 se repeated division; remainders record karo; unhe bottom-to-top padhna.

1 3_{10}

ko binary mein convert karo.

1101

2 0_{10}

ko binary mein convert karo.

10100

Repeated division mein most significant bit kaunsa remainder hota hai?

AAKHRI remainder (jab quotient 0 ho jaata hai).

n

bits kitni distinct values represent kar sakta hai, aur sabse badi value kya hai?

2^{n}

values,

0

2^{n} - 1

tak.

8 bits mein sabse badi value?

1111111 1_{2} = 255

Counting mein har bit apne right wale bit ke relative kitni baar flip karta hai?

Aadha baar (uska weight double hai).

N \div 2

ka remainder aakhri binary bit kyun deta hai?

Ye 1 hai iff

N

odd hai;

2^{0}

ek maatra odd-weighted column hai.

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho light switches ki ek row hai. Har switch sirf OFF ya ON ho sakta hai — bas itna hi kar sakta hai. Hum kehte hain OFF matlab 0 aur ON matlab 1. Count karne ke liye, sabse right wala switch har step mein flip karta hai. Jab wo aur upar nahi ja sakta (already ON hai), to OFF ho jaata hai aur left wale switch ko flip karne ke liye "dhakka" deta hai — bilkul car ke mileage counter ki tarah jahan 9 roll over ho kar 0 ho jaata hai aur agla digit bump ho jaata hai. Switches ki ek row ka normal number mein matlab nikaalney ke liye, har switch ko ek value do: right wala 1 worth hai, phir 2, 4, 8, 16… (har baar double). Sirf jo switches ON hain unki values add karo. Bas, wahi number hai!

Connections

Place-value and number bases — general $\sum d_{i} b^{i}$ idea (binary $b = 2$ ka case hai).
Hexadecimal and Octal — lambi binary strings likhne ke shortcuts (base 16, base 8).
Bits and Bytes — ek byte = 8 bits = values 0–255.
Logic gates and switches — hardware two-state kyun hai.
Binary addition and overflow — counting/carry ko arithmetic tak generalize karna.
Two's complement — negative numbers in bits ko reuse kaise karte hain.