2. Number Bases

Video coming soon2 worked examples

Theory

A number written in base $b$ uses digits $0$ to $b-1$ , with place values that are powers of $b$ . To convert to base 10, multiply each digit by its place value and add. To convert from base 10, divide repeatedly by $b$ and read the remainders from the bottom up.

The Golden Rule: converting to base 10 uses place values (powers of the base); converting from base 10 uses repeated division, reading the remainders in reverse.

⚠️ Common Examiner Traps

Remainder order: when dividing down, the base- $b$ digits are read from the last remainder to the first.
Place values: the rightmost digit has place value $b^0 = 1$ , then $b^1, b^2, \dots$ moving left.
Digit range: every digit must be less than the base.

Worked examples

Example 1

Convert $2143_{5}$ (base 5) to base 10.

Step 1: Write each digit against its place value (powers of 5):

2143_{5} = 2(5^3) + 1(5^2) + 4(5^1) + 3(5^0)

Step 2: Evaluate and add:

= 250 + 25 + 20 + 3 = 298

Example 2

Convert $100$ (base 10) to base 7.

Step 1: Divide repeatedly by 7, recording each remainder:

\begin{aligned} 100 \div 7 &= 14 \ \text{r } 2 \\ 14 \div 7 &= 2 \ \text{r } 0 \\ 2 \div 7 &= 0 \ \text{r } 2 \end{aligned}

Step 2: Read the remainders from the bottom up:

100 = 202_{7}