How to Convert Binary, Denary and Hexadecimal by Hand

To convert between binary, denary and hexadecimal by hand, use place values for conversions involving denary and groups of four binary digits for conversions between binary and hexadecimal. Binary is base $2$, denary is base $10$, and hexadecimal is base $16$. A base tells you how many different digit values a number system uses; a subscript shows which base a written number belongs to.

Binary and denary

Each binary position, reading from right to left, is worth twice the previous one: $1,2,4,8,16,32,64,128$, and so on. Add the values wherever the binary digit is $1$. For example:

$$10101101_2=128+32+8+4+1=173_{10}$$

To go from denary to binary, work through the place values from largest to smallest. For $173$, take $128$, leaving $45$; take $32$, leaving $13$; then take $8$, $4$ and $1$. Put $1$ in those positions and $0$ in the others to get $10101101_2$. Check by adding the values marked $1$. A $0$ contributes nothing, but its position still matters: $101_2$ and $11_2$ have different values.

Binary and hexadecimal

One hexadecimal digit represents exactly four binary digits, called a nibble. Group binary digits in fours from the right, adding zeros on the left if needed. Then convert each group separately:

$$10101101_2={\mathrm{AD}}_{16}$$

Here, $1010_2$ is $10$, written $\mathrm{A}$ in hexadecimal, and $1101_2$ is $13$, written $\mathrm{D}$. To convert back, replace each hexadecimal digit with its four-digit binary equivalent: $\mathrm{A}=1010$ and $\mathrm{D}=1101$. If the leftmost group has fewer than four digits, pad only its left side. Leading zeros do not change the number's value.

Denary and hexadecimal

Hexadecimal uses $\mathrm{A}$ to $\mathrm{F}$ for values $10$ to $15$. To convert $173$ to hexadecimal, divide by $16$: the quotient is $10$ and the remainder is $13$. These become $\mathrm{A}$ and $\mathrm{D}$, so $173_{10}={\mathrm{AD}}_{16}$. Check in reverse using hexadecimal place values:

$${\mathrm{AD}}_{16}=10\times 16+13=173_{10}$$

Common mistake: group binary digits from the right, not the left. For denary-to-binary conversion by repeated division, read the remainders from bottom to top.