How Do You Convert Binary to Decimal and Hexadecimal?

To convert binary to decimal, multiply each bit by its power-of-two place value and add the results. To convert binary to hexadecimal, divide the binary number into four-bit groups and replace each group with its hexadecimal digit.

Binary to decimal

Binary is a base-$2$ number system using positional notation. Starting from the right, the place values are $2^0, 2^1, 2^2, 2^3$, and so on.

The rightmost bit is the least significant bit (LSB), while the leftmost displayed bit is the most significant bit (MSB). A bit set to $1$ contributes its place value; a bit set to $0$ contributes nothing. Therefore, only powers corresponding to $1$ bits appear in the final sum.

For example, convert $11010110_2$ to decimal:

$$(1 \times 2^7) + (1 \times 2^6) + (0 \times 2^5) + (1 \times 2^4) + (0 \times 2^3) + (1 \times 2^2) + (1 \times 2^1) + (0 \times 2^0)$$ $$128 + 64 + 16 + 4 + 2 = 214$$

Therefore, $11010110_2 = 214_{10}$.

Binary to hexadecimal

Each hexadecimal digit represents one nibble, meaning four binary bits. Group the bits from the right:

$$1101\ 0110$$

Hexadecimal is base $16$ and uses digits $0$ to $9$ and letters A to F, where A represents $10$ and F represents $15$. Four bits map exactly to one hexadecimal digit because they provide $16$ possible patterns. Hexadecimal therefore displays long binary values more compactly.

Thus, $11010110_2 = \mathrm{D6}_{16}$. This can be checked using hexadecimal place values: $(13 \times 16) + 6 = 214$.

A common misconception is to form four-bit groups from the left. Always group from the least significant bit on the right; if necessary, add leading zeros on the left. For example, $10110_2$ becomes $0001\ 0110_2$, not $1011\ 0000_2$. For a bit string whose length is not a multiple of four, leading zero padding preserves its value.

IB exam technique

In A1.2 Data representation and computer logic, show either the power-of-two place values or the four-bit groupings. Examiners should be able to follow your conversion method; retaining leading zeros within each nibble also prevents avoidable hexadecimal errors. Label the bases and keep every intermediate group visible; an unsupported final answer may lose method marks if an error occurs.