Binary System Explained

Conversions, Addition, Subtraction & More

Why Binary Still Rules the Digital World

Open any smartphone, game console, or smart toaster and you'll find the same language humming underneath: binary, a two-symbol code of 0s and 1s that electronics can spot faster than you can blink. It's not a mysterious "computer math"—it's the simplest way to build reliable circuits.

• Only two states to detect. A wire is either energized (1) or not (0).
• Easy to combine. Logic gates snap together like Lego, producing every higher-level function from adders to AI chips.
• Resilient to noise. A small voltage wobble won't confuse a circuit that merely asks, high or low?

If engineers had chosen decimal wiring, every transistor would need to recognize ten states, turning a sleek laptop into a heat-spewing brick. Two states keep hardware—and your energy bill—lean.

The Base-2 Mindset

Binary is just another positional system, like decimal. The difference? Decimal digits sit in "powers of 10" columns, while binary digits live in "powers of 2" columns.

Decimal Place Values

Decimal Place	Value
10³	1,000
10²	100
10¹	10
10⁰	1

Binary Place Values

Binary Place	Value
2³	8
2²	4
2¹	2
2⁰	1

Write 10010₂ (that little subscript "2" means "base 2") and you're saying:

1⋅2⁴ + 0⋅2³ + 0⋅2² + 1⋅2¹ + 0⋅2⁰ = 16 + 0 + 0 + 2 + 0 = 18₁₀

Fast Decimal ↔ Binary Conversions

Decimal → Binary in Five Steps

1. Find the largest power of 2 less than or equal to the number.
2. Write a 1 in that column.
3. Subtract and repeat for the remainder.
4. Fill leftover columns with 0s.
5. Read the 1s and 0s left to right.

Binary → Decimal in Two Lines

1. List the powers of 2 under each 1.
2. Add them up.

Binary Arithmetic Essentials

Binary Addition Without Tears

Binary addition mirrors grade-school column addition; you just "carry" when 1 + 1 hits 2 (which is "10₂").

Addends	Carry	Sum
0 + 0	0	0
0 + 1 or 1 + 0	0	1
1 + 1	1	0

Binary Subtraction: Borrowing 101

Borrowing happens only when you try to subtract 1 from 0.

Operation	Borrow?	Result
0 − 0	–	0
1 − 0	–	1
1 − 1	–	0
0 − 1	Borrow 1	1

Borrowing turns that lone "0" into "2" (10₂) before subtracting. Cascade borrowing works exactly like decimal long subtraction.

Advanced Binary Concepts

Signed Numbers and Two’s Complement

Computers must store negatives, too. The most popular trick is two's complement:

1. Write the positive binary
2. Invert every bit (0 → 1, 1 → 0)
3. Add 1

Real-World Binary Applications

Industry	Binary Use Case	Why It Matters
Networking	IP addresses	Routing decisions hinge on bit-level masks
Multimedia	JPEG & MP3 compression	Bit patterns flag "start of frame"
Cryptography	AES, RSA algorithms	Key operations use binary mod-exp calculations
Finance	High-frequency trading	FPGA accelerators process 64-bit integers
Spacecraft	Telemetry packets	Compact bitstreams travel millions of miles

Practice Problems

Key Takeaways