Binary / Base 2

Binary, also called Base 2, is a number system that uses only two digits: 0 and 1. Every number you normally write in decimal (base 10) can also be written in binary using just these two digits.

Binary is a positional numeral system with a base (or radix) of 2. Each digit in a binary number represents a successive power of 2, starting from $2^0$ at the rightmost position. A binary number $d_n d_{n-1} \ldots d_1 d_0$ (where each $d_i$ is either 0 or 1) has the decimal value $d_n \times 2^n + d_{n-1} \times 2^{n-1} + \cdots + d_1 \times 2^1 + d_0 \times 2^0$ .

Key Formula

\text{Value} = d_n \times 2^n + d_{n-1} \times 2^{n-1} + \cdots + d_1 \times 2^1 + d_0 \times 2^0

Where:

$d_i$ = the digit (0 or 1) at position i, counting from the right starting at 0
$n$ = the position of the leftmost digit

Worked Example

Problem: Convert the binary number 110101 to decimal (base 10).

Step 1: Write out the place values. Starting from the right, each position represents a power of 2.

2^5,\; 2^4,\; 2^3,\; 2^2,\; 2^1,\; 2^0 = 32,\; 16,\; 8,\; 4,\; 2,\; 1

Step 2: Line up each digit of 110101 with its place value and multiply.

1 \times 32,\; 1 \times 16,\; 0 \times 8,\; 1 \times 4,\; 0 \times 2,\; 1 \times 1

Step 3: Evaluate each product.

32,\; 16,\; 0,\; 4,\; 0,\; 1

Step 4: Add the results together to get the decimal value.

32 + 16 + 0 + 4 + 0 + 1 = 53

Answer: The binary number 110101 equals 53 in decimal.

Visualization

Why It Matters

Binary is the language of computers. Every piece of data a computer processes — text, images, music, video — is stored and manipulated as sequences of 0s and 1s. This is because electronic circuits have two natural states: on (1) and off (0). Understanding binary helps you see how digital technology works at its most fundamental level.

Common Mistakes

Mistake: Using powers of 10 instead of powers of 2 when converting binary to decimal.

Correction: In base 2, the place values are 1, 2, 4, 8, 16, 32, … (powers of 2), not 1, 10, 100, 1000, … (powers of 10). Always multiply each binary digit by the corresponding power of 2.

Mistake: Reading binary digits from left to right starting with

2^1

instead of assigning powers from the right.

Correction: Powers of 2 are assigned starting from the rightmost digit as

2^0

, then increasing leftward. The rightmost digit is always the ones place (

2^0 = 1

Related Terms

Base — Binary is a specific base (base 2)
Digit — Binary uses only the digits 0 and 1
Number Line — Binary numbers can be placed on a number line