Binary Number System Basics

Binary Basics

• Computers rely on millions of micro-transistors with only two states: on / off → represented by $1$ / $0$.
• Binary is a base-$2$ number system; every value stored, transmitted, or processed is reduced to sequences of $0$s and $1$s.

Bits & Bytes

• Bit: single binary digit.
• Byte: $8$ bits → standard unit for data (e.g.
  $\text{Byte}=\underbrace{8\,\text{bits}}<em>{b</em>7\,b<em>6\,b</em>5\,b<em>4\,b</em>3\,b<em>2\,b</em>1\,b_0}$).

Place Values in Binary (one byte)

• Powers of $2$, right → left: $2^0=1,\;2^1=2,\;2^2=4,\;2^3=8,\;2^4=16,\;2^5=32,\;2^6=64,\;2^7=128$.

Denary (Decimal) vs Binary

• Decimal uses powers of $10$; binary uses powers of $2$.
• Example conversion: $10011010<em>2 = 128+16+8+2 = 154</em>{10}$.

Converting Binary → Denary

• Write the value under each $1$ bit, then add.
• Quick rule: sum of all powers of $2$ where bit = $1$.

Converting Denary → Binary

Method 1 – Place-value / “coin” subtraction

• Start with highest power of $2$ ≤ number.
• Subtract, mark $1$; continue until remainder $0$.

Method 2 – Successive division by $2$

• Repeatedly divide the number by $2$, recording remainders.
• Read remainders bottom-to-top to form binary.

Example (both methods):

• Convert $207<em>{10}$. • Largest power $128$ → remainder $79$. • Next $64$ → remainder $15$. • $8,4,2,1$ all fit. → $207</em>{10}=11001111_2$.

Key Takeaways

• Binary’s simplicity matches transistor states, providing speed & reliability.
• 1 bit = 2 states, 8 bits = 1 byte (256 possible values $0!\text{–}!255$).
• Mastery of place values and two conversion methods covers all basic exam tasks.