Place value is a system of powers
A number base tells you how many distinct digits are used and how the value of a position changes as you move left or right. In base ten, each position is worth ten times the position immediately to its right. Thus 4,307 means 4 × 10³ + 3 × 10² + 0 × 10 + 7. The written digits describe a quantity through their positions. In base five, the same idea uses powers of five, and the permitted digits are 0, 1, 2, 3 and 4.
A subscript identifies the base: 132₅ means one group of 25, three groups of 5 and two units. Its decimal value is 25 + 15 + 2 = 42. The expression 152₅ is not a valid base-five numeral because digit 5 is unavailable. In any base b, digits run from 0 to b − 1. Zero remains important even though it adds no quantity at a position: 101₂ is different from 11₂ because the zero preserves the missing twos position.
Convert a whole number into base ten
Write the digits against descending powers of the base, multiply and add. For 203₄, calculate 2 × 4² + 0 × 4 + 3 = 35. For 11010₂, calculate 1 × 16 + 1 × 8 + 0 × 4 + 1 × 2 + 0 = 26. Start with the rightmost whole-number position as zero. A useful check is that an n-digit positive number in base b lies between bⁿ⁻¹ and bⁿ − 1. For example, every three-digit binary number is between 4 and 7 in decimal.
Convert from base ten by repeated division
To express 47 in base five, divide by 5 and record remainders. The first division gives quotient 9 remainder 2. Dividing 9 gives quotient 1 remainder 4. Dividing 1 gives quotient 0 remainder 1. Read the remainders from last to first: 142₅. Checking gives 25 + 20 + 2 = 47. The first remainder is the units digit, so reading it first would reverse the required order. Stop once the quotient is zero.
When converting between two nondecimal bases, converting through decimal is a dependable general method. To convert 23₄ into binary, first obtain 2 × 4 + 3 = 11, then express 11 as 8 + 2 + 1 to get 1011₂. Bases that are powers of each other offer a shortcut. One base-four digit corresponds to two binary digits: 0 ↔ 00, 1 ↔ 01, 2 ↔ 10 and 3 ↔ 11. Therefore 23₄ becomes 10 11₂, the same answer.
Add, subtract and multiply within a base
Arithmetic uses the base as the carrying threshold. In base. To add 1011₂ and 0110₂, work from the right: 1 + 0 = 1; next, writing 0 and carrying 1; then 0 + 1 + carried; finally 1 + 0 + carried. The result is 10001₂. Decimal checking gives 11 + 6 = 17.
Borrowing works similarly. In base five, borrowing one unit from the next column contributes five units to the column. For 42₅ − 14₅, borrow from the fours column: the units calculation becomes 7 − 4 = 3 and the next column becomes 3 − 1 = 2, giving 23₅. This agrees with 22 − 9 = 13 in decimal. In binary multiplication, multiply by either 0 or 1 and shift each successive row one binary place, just as decimal long multiplication shifts by powers of ten.
Fractional positions use negative powers
Positions after the radix point represent b⁻¹, b⁻² and so on. Thus 0.101₂ = 1/2 + 0/4 + 1/8 = 5/8 = 0.625 in decimal. To convert a decimal fraction into another base, repeatedly multiply the fractional remainder by the new base and record each integer part in order. For 0.625 in binary, multiplying by 2 gives 1.25, recording 1 and retaining 0.25. Next comes 0.5, recording 0. The final multiplication gives 1.0, recording 1 and leaving no remainder.
Not every terminating decimal has a terminating expansion in every base. For example, 0.1 in decimal repeats in binary. If a question requests a fixed number of base digits, distinguish a truncation from rounding and state the accuracy used. To check a fractional conversion, return to the sum of negative powers rather than treating the digits after the point as an ordinary decimal fraction. The symbol 0.2₅ equals 2/5, not 2/10.
Find an unknown base and check your answer
If 23ᵦ has decimal value 17, then 2b + 3 = 17 and b = 7. Check the admissibility condition b > 3 because digit 3 occurs. An algebraic solution smaller than or equal to the largest digit cannot be the base. Base problems test place-value reasoning, not just conversion routines. A numeral may represent different quantities in different bases, but the underlying quantity remains unchanged when its representation is converted correctly.
Self-check: convert 10101₂ to decimal, express decimal 19 in base three, and evaluate 0.11₂. The answers are 21, 201₃ and 3/4. For the second. Avoid mixing a base-ten carry with a nondecimal addition, omitting an interior zero, or introducing a digit equal to the base. Write the base on the final answer, because the same string of digits without its base can be ambiguous.
Explain why some fractional expansions terminate
A fraction terminates in base b when, after reducing the fraction, its denominator divides a of b. For example, 1/4 terminates in base two because, giving 0.01₂. The fraction 1/5 terminates in base five as 0.1₅ but does not terminate in base two. This is a property of the representation, not a change in the fraction's value. If a conversion is stopped after several places, write an approximation sign rather than an exact equality when a nonzero remainder remains.
Division can be checked through multiplication in the same base. The quotient of 10100₂ by 100₂ is 101₂ because. A shift of two binary places multiplies by 2² = 4, while a shift of two decimal places multiplies by 100. Keeping the base attached to the operation prevents applying a familiar decimal shortcut with the wrong scale. For any unfamiliar calculation, a decimal conversion is a useful verification, but show the requested nondecimal working when the assessment asks specifically for arithmetic in that base.
As a final check, ask whether every digit is smaller than the base and whether an answer of the expected size has been obtained. A sum of two positive quantities cannot be smaller than either addend, regardless of the chosen numeral system.