A number base tells us the value of each digit position. In base $b$, the place values from the right are $b^0, b^1, b^2, \dots$. Base 10 is the everyday system; SPM also uses base 2, base 5 and base 8.
Converting to base 10
Multiply each digit by its place value and add. For example $1011_2 = 1\times2^3+0\times2^2+1\times2^1+1\times2^0 = 8+0+2+1 = 11$. Similarly $23_5 = 2\times5+3 = 13$.
Key formula
Value of a digit $=$ digit $\times b^{\text{position}}$, counting positions $0,1,2,\dots$ from the right in base $b$.
Converting from base 10
Repeatedly divide by the new base and record the remainders. Reading the remainders from bottom to top gives the answer.
Worked example
Convert $13_{10}$ to base 2. $13\div2=6$ r $1$; $6\div2=3$ r $0$; $3\div2=1$ r $1$; $1\div2=0$ r $1$. Reading remainders bottom-up gives $1101_2$. Check: $8+4+0+1=13$.
Remember
- Base 5 uses only the digits $0$ to $4$.
- Read the division remainders from bottom to top.
- Always check by converting back to base 10.