1
Digits in Base $b$
A number in base $b$ can only contain digits strictly less than $b$:
Base 2: 0, 1
Base 5: 0, 1, 2, 3, 4
Base 8: 0, 1, 2, 3, 4, 5, 6, 7
2
Place Value Expansion
To convert any base $b$ into Base 10, multiply each digit by its positional power $b^k$:
$$1101_2 = (1 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 13_{10}$$
3
Repeated Division
To convert Base 10 into Base $n$, repeatedly divide by $n$ until quotient is 0. Read remainders from bottom to top:
$$43_{10} \div 5 \implies \text{Remainders: } 1, 3, 1 \implies 133_5$$
SPM Fast-Track Shortcut: Base 2 to Base 8 Direct Grouping
Because $2^3 = 8$, you can convert between Base 2 and Base 8 directly by grouping binary digits into triplets (groups of 3) from right to left!
$$\underbrace{101}_{5} \quad \underbrace{011}_{3} \quad \underbrace{110_2}_{6} = 536_8$$