Chapter
Number Bases
Base conversion, arithmetic in non-decimal bases, binary notation, digit counts, and trailing zeros in other bases.
Theory
1. Place Value
In base \(b\), the numeral \(a_ka_{k-1}\cdots a_0\) represents \(a_kb^k+a_{k-1}b^{k-1}+\cdots+a_0\).
2. Converting and Arithmetic
Repeated division by the base gives digits from right to left. Arithmetic works as usual, but carrying happens at the base.
3. Trailing Zeros
Trailing zeros in base \(b\) are controlled by the prime factorization of \(b\).
Examples
Example 1. From Base Seven
This is the basic place-value conversion.
Problem.
Convert \(345_7\) to decimal notation.
Solution.
\(345_7=3\cdot7^2+4\cdot7+5=147+28+5=180\).
Example 2. To Base Five
This shows conversion from decimal by powers of the base.
Problem.
Write \(2026\) in base \(5\).
Solution.
\(2026=3\cdot625+1\cdot125+1\cdot25+0\cdot5+1\), so \(2026=31101_5\).
Example 3. Addition in Base Five
This adds arithmetic in non-decimal bases.
Problem.
Compute \(234_5+143_5\) and write the answer in base \(5\).
Solution.
\(234_5=69\) and \(143_5=48\), so the sum is \(117=432_5\).
Example 4. Trailing Zeros in Base Twelve
This connects bases with prime factorization.
Problem.
How many zeros does \(10!\) end with when written in base \(12\)?
Solution.
In \(10!\), \(v_2=8\) and \(v_3=4\). Each \(12\) uses two factors \(2\) and one factor \(3\), so the answer is \(\min(\lfloor8/2\rfloor,4)=4\).
Problems
Problems
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Ladders
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