Q.Expand $\log_b\left(\dfrac{a^{a} b^{b}}{c^{c} d^{d}}\right)$.

Binary Numbers: Counting with Only Two Fingers

Imagine you had only two fingers on each hand — one up, one down. How would you count? You'd still manage, but every number would be built from just two symbols: 0 and 1. That's exactly what binary is: a number system that uses only two digits.

You already know the decimal system (base 10), where each position represents a power of 10: units (10⁰), tens (10¹), hundreds (10²), and so on. Binary (base 2) works the same way, but each position represents a power of 2: 1 (2⁰), 2 (2¹), 4 (2²), 8 (2³), 16 (2⁴), etc.

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How Binary Counting Works

In decimal, when you reach 9, the next number wraps around to 10 — you add a new place. In binary, you wrap around at 1. So counting goes:

Notice the pattern: every time you run out of digits (reach 1), you add a new place to the left, just like decimal does at 9.

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Converting Binary to Decimal

Take the binary number 1101. Write the place values above each bit (starting from the rightmost as 2⁰):

Place:  8   4   2   1
Bits:   1   1   0   1

Now add the values where there's a 1: $8 + 4 + 0 + 1 = 13$.

So $1101_2 = 13_{10}$.

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Converting Decimal to Binary

To convert 13 back to binary, repeatedly divide by 2 and record the remainders:

1. $13 \div 2 = 6$ remainder 1 (rightmost bit)
2. $6 \div 2 = 3$ remainder 0
3. $3 \div 2 = 1$ remainder 1
4. $1 \div 2 = 0$ remainder 1 (leftmost bit)

Read the remainders from bottom to top: 1101. That's your binary number.

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Why Binary Matters

Computers use binary because transistors (the building blocks of processors) have two natural states: on (1) and off (0). Every piece of data — numbers, text, images, sound — is ultimately stored as sequences of bits. A single 0 or 1 is a bit; 8 bits make a byte; 1024 bytes make a kilobyte, and so on.

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A Final Intuition

Think of binary as a light switch. Each switch can be off (0) or on (1). With one switch, you can represent 2 states (0 or 1). With two switches, 4 states (00, 01, 10, 11). With $n$ switches, $2^n$ states. That exponential growth is why computers can handle such vast amounts of information with just two symbols.

Binary numbers are the base-2 representation of numbers using only the digits 0 and 1, where each digit's place value is a power of 2.