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

WHY binary at all?

WHAT is place-value (the engine behind every base)

Derivation from scratch (decimal first, so the idea is familiar): Why does "253" equal two-hundred-fifty-three? Because we agreed each slot is worth ten times the slot to its right: $253 = 2\cdot 10^2 + 5\cdot 10^1 + 3\cdot 10^0 = 200+50+3$ Why this step? The leftmost digit is "further from the units" by two positions, so it's scaled by $10^2$ .

Now swap base 10 → base 2. The only thing that changes is the weights become powers of 2, and allowed digits shrink to $\{0,1\}$ :

HOW to convert Binary → Decimal

Worked example Convert

$1011_2$ to decimal Write weights above the bits:

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

Add up the weights where the bit is 1: $8 + 0 + 2 + 1 = 11$ Why this step? A 0 contributes nothing; a 1 contributes its place weight. So you just sum the "on" columns. Answer: $1011_2 = 11_{10}$ .

HOW to convert Decimal → Binary (repeated division)

Counting in binary

Worked example Count 0 to 7 in 3 bits

Why this step? Watch the rightmost bit flip every step (1,0,1,0…), the middle bit flip every 2 steps, the left bit every 4 steps — each column flips half as often as the one to its right, because its weight is double.

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

Forecast-then-Verify

Recall Forecast: how many bits to write

$100_{10}$ ? Then verify. Forecast: $2^6=64$ , $2^7=128$ . Since $64 \le 100 < 128$ , we need 7 bits. Verify (repeated division): $100\to50$ r0, $50\to25$ r0, $25\to12$ r1, $12\to6$ r0, $6\to3$ r0, $3\to1$ r1, $1\to0$ r1 → bottom-up $1100100$ . That's 7 digits ✓ and $64+32+4=100$ ✓.

Common mistakes

Flashcards

What base does the binary system use, and why is it natural for computers?

Base 2; computer switches have exactly two stable states (off=0, on=1), which are cheap and noise-resistant.

Binary→decimal: what is the weight of the bit at position

i

(rightmost = 0)?

2^i

Convert

1011_2

to decimal.

8+0+2+1 = 11

Convert

110100_2

to decimal.

32+16+4 = 52

What algorithm converts decimal to binary?

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

Convert

13_{10}

to binary.

1101

Convert

20_{10}

to binary.

10100

In repeated division, which remainder is the most significant bit?

The LAST remainder (when quotient reaches 0).

How many distinct values can

n

bits represent, and what is the largest?

2^n

values, from

0

2^n-1

Largest value in 8 bits?

11111111_2 = 255

In counting, how often does each bit flip relative to the bit on its right?

Half as often (its weight is double).

Why does the remainder of

N\div2

give the last binary bit?

It's 1 iff

N

is odd;

2^0

is the only odd-weighted column.

Recall Feynman: explain to a 12-year-old

Imagine a row of light switches. Each switch is OFF or ON — that's all it can do. We say OFF means 0 and ON means 1. To count, the rightmost switch flips on every step. When it can't go higher (it's already ON), it turns OFF and "pokes" the switch to its left to flip — just like a car's mileage counter where 9 rolls over to 0 and bumps the next digit. To find what a row of switches means as a normal number, give each switch a value: the right one is worth 1, then 2, 4, 8, 16… (doubling each time). Add up the values of only the switches that are ON. That's the number!

Connections

Place-value and number bases — the general $\sum d_i b^i$ idea (binary is the $b=2$ case).
Hexadecimal and Octal — shortcuts for writing long binary strings (base 16, base 8).
Bits and Bytes — a byte = 8 bits = values 0–255.
Logic gates and switches — why hardware is two-state.
Binary addition and overflow — counting/carry generalized to arithmetic.
Two's complement — how negative numbers reuse these bits.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, computer ke andar sirf switches hote hain — ya toh OFF (0) ya ON (1). Bas do hi states. Isiliye computer ka apna number system base 2 yaani binary hai. Decimal (base 10) hum isliye use karte hain kyunki humare paas 10 ungliyaan hain, warna usme bhi koi magic nahi hai. Dono mein same "place-value" ka funda chalta hai: har position ka ek weight hota hai. Decimal mein weights hote hain 1, 10, 100… (das se multiply); binary mein weights hote hain 1, 2, 4, 8, 16… (do se multiply, har baar double).

Binary se decimal nikalna sabse easy hai: bit ke upar uska weight likho ( $2^0, 2^1, 2^2 \dots$ right se left), aur jahan-jahan bit 1 hai sirf wahi weights jod do. Jaise $1011$ → $8+0+2+1 = 11$ . Bas itna hi!

Decimal se binary ke liye "repeated division by 2" karo: number ko 2 se divide karte raho, har baar remainder (0 ya 1) note karo, aur quotient ko aage le jao jab tak 0 na aa jaye. Phir remainders ko neeche se upar padho — kyunki pehla remainder units wala bit hai (sabse chhota), aur aakhri remainder sabse bada (leftmost) bit. $13$ ko karo toh $1101$ aata hai, check: $8+4+1=13$ .

Counting toh bilkul gaadi ke meter jaisa hai — rightmost digit har step pe flip hota hai, jab woh ON se aage nahi ja sakta toh 0 ho ke left wale ko carry deta hai. Aur ek important baat: $n$ bits se aap $2^n$ alag values bana sakte ho, lekin sabse bada number $2^n-1$ hota hai (kyunki ginti 0 se shuru hoti hai). 8 bits = 256 values, max = 255. Yeh off-by-one wali galti exam mein bahut log karte hain, isse bach ke rehna!

$N$	$N \div 2$	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

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

WHY binary at all?

WHAT is place-value (the engine behind every base)

HOW to convert Binary → Decimal

HOW to convert Decimal → Binary (repeated division)

Counting in binary

Forecast-then-Verify

Common mistakes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — How Computers Work

Connections