Mixed practice: A Story of Numbers
A Story of Numbers24 min readLesson 6 of 8
6.1 Conversions between bases
Question 1. Write 53 in base 2, base 4, base 5 and base 7.
Answer. Base 2: 53 = 32 + 16 + 4 + 1 = 110101. Base 4: 53 = 3 × 16 + 1 × 4 + 1 = 311. Base 5: 53 = 2 × 25 + 0 × 5 + 3 = 203. Base 7: 53 = 1 × 49 + 0 × 7 + 4 = 104.
Question 2. Convert 1101 in base 2, 231 in base 4, 304 in base 5 and 1001 in base 7 to decimal.
Answer. 1101 = 8 + 4 + 0 + 1 = 13. 231 = 2 × 16 + 3 × 4 + 1 = 45. 304 = 3 × 25 + 0 × 5 + 4 = 79. 1001 = 1 × 343 + 0 + 0 + 1 = 344.
Question 3. List the landmarks of base 6 up to four places.
Answer. The landmarks are 6^0 = 1, 6, 36 and 216.
Question 4. Write 100 in base 8.
Answer. 100 = 1 × 64 + 4 × 8 + 4, so the answer is 144 in base 8.
Question 5. Which is larger: the numeral 1234 read in base 6 or the same digits read in base 8?
Answer. The base-8 value is larger. In base 6, 1234 = 1 × 216 + 2 × 36 + 3 × 6 + 4 = 310; in base 8, 1234 = 1 × 512 + 2 × 64 + 3 × 8 + 4 = 668. Larger place values make the same digits worth more.
6.2 Roman numerals
Question 6. Write 1489 in Roman numerals.
Answer. 1489 = 1000 + 400 + 80 + 9 = M + CD + LXXX + IX = MCDLXXXIX.
Question 7. Read MCMXCIV.
Answer. M = 1000, CM = 900, XC = 90 and IV = 4, so the value is 1994.
6.3 Egyptian numerals
Question 8. Represent 3270 in Egyptian numerals and say how many of each symbol are needed.
Answer. 3270 = 3 × 1000 + 2 × 100 + 7 × 10. The numeral uses three 1000-symbols, two 100-symbols and seven 10-symbols, with no unit symbols.
Question 9. An Egyptian numeral shows four 1000-symbols, five 10-symbols and three unit symbols. What is its value?
Answer. 4 × 1000 + 5 × 10 + 3 = 4053.
6.4 Mayan numerals
Question 10. Write 88 in the Mayan system.
Answer. 88 = 4 × 20 + 8, so the numeral has four dots in the 20s place and, in the units place, one bar for 5 plus three dots for 3.
Question 11. A Mayan numeral has one dot in the 360s place, a shell in the 20s place and one bar in the units place. Find its value.
Answer. 1 × 360 + 0 × 20 + 5 = 365.
6.5 Chinese rods and Mesopotamian numerals
Question 12. Write 908 in Chinese rod numerals and explain the blank place.
Answer. 908 = 9 hundreds + 0 tens + 8 units. The hundreds rod shows 9, the tens place is left blank because there are no tens, and the units rod shows 8. The blank must be kept, or the numeral would be read as 98.
Question 13. Write 3725 in the Mesopotamian base-60 system.
Answer. 3725 = 1 × 3600 + 2 × 60 + 5, so the coefficients are 1, 2 and 5 from the highest place downwards.
Question 14. Which system needs fewer symbols for very large numbers: Roman or base-60? Explain.
Answer. Base-60 needs fewer symbols, because its place values grow by a factor of 60 each time, while Roman numerals need new symbols and long strings as the number grows.
6.6 Reading and checking
Question 15. Write 3999 in Roman numerals and show the regroupings.
Answer. 3999 = 3000 + 900 + 90 + 9 = MMM + CM + XC + IX = MMMCMXCIX. Each subtractive pair handles one landmark below the next full symbol.
Question 16. An Egyptian numeral contains twelve 10-symbols. Why is that not a finished numeral, and what should replace them?
Answer. Ten of the 10-symbols make 100, so they regroup into one 100-symbol. That leaves two 10-symbols and one 100-symbol. A correct numeral never repeats a symbol ten times, because regrouping always produces the next landmark.
Question 17. Write 400 in Mayan numerals and explain each place.
Answer. 400 = 1 × 360 + 2 × 20 + 0. The numeral has one dot in the 360s place, two dots in the 20s place and a shell in the units place. The shell is needed so the 20s place is not mistaken for the units place.
Question 18. A Chinese rod numeral has a 9 in the hundreds place, a blank in the tens place and a 6 in the units place. What number is it, and what would be read if the blank were ignored?
Answer. The number is 906. If the blank were ignored, the numeral would be read as 96, because the 6 would move into the tens place.
6.7 Method notes
For each conversion, write the landmark powers first, then take the largest landmark that fits and continue with the remainder. For reading a positional numeral, multiply each digit by its place's value and add. For Roman numerals, work from the largest landmark down and use the subtractive forms for 4, 9, 40 and 90. For Egyptian numerals, count symbols by landmark and regroup ten of a kind into the next landmark. For Mayan numerals, remember that the third place is 360, not 400. For Chinese rods, never drop a blank place. These habits turn a gallery of systems into a single reliable method.