Binary To Decimal Converter

A binary to decimal conversion rewrites a base-$2$ number as a base-$10$ number. The key idea is simple: each binary digit tells you whether to include a power of $2$. A $1$ means include that place value. A $0$ means skip it.

For example, $1011_2$ equals $11_{10}$ because it includes $8$, skips $4$, includes $2$, and includes $1$.

How Binary Place Value Turns Into Decimal

Binary is a base-2 system, so its place values are powers of $2$ rather than powers of $10$. From right to left, the places are:

$$2^0,\; 2^1,\; 2^2,\; 2^3,\; \dots$$

That means the first few place values are:

$$1,\; 2,\; 4,\; 8,\; 16,\; \dots$$

If a digit is $1$, that place value counts. If a digit is $0$, it does not.

The Rule Behind Binary To Decimal Conversion

For a binary number with digits $b_n b_{n-1} \dots b_1 b_0$, where each $b_i$ is either $0$ or $1$, the decimal value is

$$\sum_{i=0}^{n} b_i 2^i$$

You do not need the formula to do the conversion, but it shows the idea clearly: binary is just place value with powers of $2$.

Worked Example: Convert $11001_2$

Start from the right, where the place values are $1, 2, 4, 8, 16$.

$$11001_2 = 1 \cdot 16 + 1 \cdot 8 + 0 \cdot 4 + 0 \cdot 2 + 1 \cdot 1$$

Now keep only the values attached to a $1$:

$$11001_2 = 16 + 8 + 1$$

So the decimal value is

$$11001_2 = 25_{10}$$

If you want a quick check, read the number from left to right as "one $16$, one $8$, zero $4$s, zero $2$s, and one $1$."

Why The Method Works

In base $10$, the number $407$ means

$$4 \cdot 10^2 + 0 \cdot 10^1 + 7 \cdot 10^0$$

Binary works the same way, but with powers of $2$:

$$11001_2 = 1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0$$

The structure is identical. Only the base changes.

Common Binary To Decimal Mistakes

1. Using powers of $10$ instead of powers of $2$. Binary place values are $1, 2, 4, 8, 16, \dots$.
2. Counting places from the left without knowing the exponent. The safest method is to start at the right with $2^0$.
3. Treating a number like $1021$ as binary. Valid binary digits are only $0$ and $1$.
4. Forgetting that leading zeros do not change the value. For example, $0011_2$ and $11_2$ both equal $3_{10}$.

When Binary To Decimal Is Used

Binary to decimal conversion appears whenever you need to interpret how computers store values. It shows up in basic computer science, digital electronics, data representation, and bit-based settings such as permissions, flags, or memory values.

Even if you never work directly with hardware, understanding binary place value makes number systems much less mysterious.

Try A Similar Conversion

Convert $101101_2$ to decimal by writing the place values first, then adding only the powers of $2$ that line up with a $1$. That one habit prevents most conversion errors.