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Class 8 A Story of Numbers: Thinking Puzzles: cheat sheet
CBSE Computational Thinking and AI. Numbers can be written in many systems: binary (base 2), base 3, base 5 and Roman numerals. Each place or letter has a fixed value.
Key ideas
- In base b the place values are 1, b, b², b³, … and each digit goes from 0 to b − 1.
- In binary, each place is double the one to its right, so any number is a sum of different powers of 2.
- Binary pipes of 1, 2, 4, 8, … inches (one of each) can make any length: the 1s in the binary number say which pipes.
- Roman numerals add letters, except a smaller one before a larger one, which is taken away.
- The fewest swaps of neighbours that sort a list equals the number of pairs that are out of order.
Remember
| Binary place values | 128, 64, 32, 16, 8, 4, 2, 1 |
| Example | 13 = 8 + 4 + 1 = 1101₂ |
| Base 3 and base 5 place values | 81, 27, 9, 3, 1 and 125, 25, 5, 1 |
| Roman numerals | I 1, V 5, X 10, L 50, C 100, D 500, M 1000 |
| Subtraction pairs | IV 4, IX 9, XL 40, XC 90, CD 400, CM 900 (49 = XLIX) |
| Largest k-digit number in base b | bᵏ − 1 (444₅ = 124) |
| Digits of n in binary | k digits when 2ᵏ⁻¹ ≤ n < 2ᵏ |
Watch out
- Reading a binary number from the wrong end: the rightmost digit is the 1s place.
- Writing 9 as VIIII or reading IX as 11.
- Using a digit as large as the base, like 3 in base 3.
Worked example
(a) With binary pipes (1, 2, 4, 8, … one of each), how many pipes measure 907 inches? (b) With decimal pipes (1, 10 and 100 inches, up to nine of each), how many pipes are needed?
Working: 907 = 1110001011 in binary (count the 1s); count the decimal digits' total: 9 + 0 + 7
Answer: (a) 6; (b) 16
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