Learn – Math circle – Base-n arithmetic

1 Conversion between labels and values

1.1 Whole numbers

1.1.1 Exercises

1.1.1.1 Problem set
  1. An island keeps currency in the form of leaves, twigs, branches, trunks. Say, 5 leaves make a twig, 5 twigs make a branch, 5 branches make a trunk. Assume Miliana has 586 leaves with her. How may she most efficiently store her money using leaves, twigs, branches and trunks?
  2. An island keeps currency in the form of grains, pebbles, rocks, boulders. Say, 7 grains make a pebble, 7 pebbles make a rock, 7 rocks make a boulder. Assume Ela has 776 grains with her. How may she most efficiently store her money using grains, pebbles, rocks and boulders?
  3. A community keeps currency in the form of petals, flowers, bouquets and garlands. Say, 9 petals make a flower, 9 flowers make a bouquet, 9 bouquets make a garland. Assume Anu has 2618 petals with her. How may she most efficiently store her money using petals, flowers, bouquets and garlands?
1.1.1.2 Problem set

In each of the following problems, give the label in required base for the value.

  1. 609 in base-6
  2. 1847 in base-7
  3. 2345 in base-10
  4. 3735 in base-6
  5. 3470 in base-4
  6. 11973 in base-5
  7. 45 in base-2
1.1.1.3 Problem set

In each of the following problems, give the value of the label.

  1. 4512_6
  2. 4134_5
  3. 3126_8
  4. 1232_4
  5. 10101010_2
1.1.1.4 Problem set

In each of the following problems, quickly give the label for the given value.

  1. 12 in base-8
  2. 12 in base-7
  3. 17 in base-9
  4. 20 in base-6
  5. 14 in base-7
  6. 18 in base-8
  7. 21 in base-5
  8. 24 in base-6
1.1.1.5 Problem set

In each of the following problems, quickly give the value of the label.

  1. 1000_9
  2. 10000_8
  3. 100000_5
1.1.1.6 Problem set

In each of the following problems, quickly give the label for the given value.

  1. 6^3 in base-6
  2. 9^5 in base-9
  3. 8^4 in base-8
  4. 7^5 in base-7
  5. 9 in base-9
  6. 6 in base-6
  7. 7 in base-7
1.1.1.7 Problem set

In each of the following problems, quickly give the label for the given value.

  1. \left(3\times 6^3 + 5\times 6^2 + 1\times 6 + 4\right) in base-6
1.1.1.8 Problem set

In each of the following problems, quickly give the label for the given value.

  1. 5^5 - 3 in base-5
  2. 7^7 - 5 in base-7
  3. 6^5 - 6 in base-6
  4. 8^4 - 11 in base-8
  5. 6\times 7^6 - 2\times 7^6 in base-7
  6. 7\times 9^3 + 6\times 9^3 in base-9
  7. 4\times 8^5 + 5\times 8^5 + 7\times 8^5 in base-8

1.2 Fractional numbers

1.2.1 Exercises

1.2.1.1 Problem set

In each of the following problems, give the value of the label.

  1. 0.453_6
  2. 401.234_5
  3. 215.402_7
1.2.1.2 Problem set

In each of the following problems, quickly give the label for the given value.

  1. \left(3\times \frac{1}{7} + 5\times \frac{1}{7^2} + 4\times \frac{1}{7^3}\right) in base-7
  2. \left(2\times \frac{1}{5} + 4\times \frac{1}{5^2} + 1\times \frac{1}{5^3}\right) in base-5
  3. \left(3\times \frac{1}{9} + 7\times \frac{1}{9^2} + 6\times \frac{1}{9^3} + 4\times\frac{1}{9^4}\right) in base-9
1.2.1.3 Problem set

In each of the following problems, give the label for the given value.

  1. 0.875 in base-6
  2. 0.5536 in base-5
  3. 0.87890625 in base-4
  4. 0.591796875 in base-8
  5. 0.234 in base-5
1.2.1.4 Problem set

In each of the following problems, give the label for the given value.

  1. 235.875 in base-6
  2. 67.8048 in base-5

2 Counting and Arithmetic

2.1 Exercises

2.1.1 Problem set

  1. Write numbers counting in base-4 up to 1000_4
  2. Write numbers counting in base-3 up to 1000_3
  3. Write numbers counting in base-2 up to 100000_2
  4. Write numbers counting in base-16 up to 25_{16}
  5. Write the last thirty-two numbers that you count in base-16 before reaching 100_{16}.
  6. Write the last thirty-two numbers that you count in base-16 before reaching 1000_{16}.

2.1.2 Problem set

For each of the following, simplify.

  1. 7_8 + 2_8
  2. 6_9 + 6_9
  3. 40_5 + 40_5
  4. 400_6 + 500_6
  5. 543_7 + 156_7

2.1.3 Problem set

For each of the following, simplify.

  1. 427_8 + 367_8
  2. 4423_6 + 1535_6
  3. 4545_6 + 1011_6
  4. 34.56_8 + 45.67_8
  5. 1111_2 + 1111_2 + 1111_2 + 1111_2

2.1.4 Problem set

For each of the following, simplify.

  1. 43_8 - 35_8
  2. 643_7 - 546_7
  3. 4240_5 - 3423_5
  4. 100001_2 - 1111_2
  5. 84.73_9 - 64.84_9

2.1.5 Problem set

For each of the following, simplify.

  1. 1011_2 \times 111_2
  2. 1111_2 \times 1000_2
  3. 1010101_2 \times 10_2
  4. 1010101_2 \div 100_2
  5. 10101010_2 \div 1000_2

3 Conversions between base-2, base-8, base-16

3.1 Exercises

3.1.1 Problem set

  1. Write 34\times 8^3+12\times8^2+23\times 8 in base-8
  2. Write 54\times 7^3+31\times 7^2+49\times 7 in base-7

3.1.2 Problem set

  1. Convert 56_8 into base-2.
  2. Convert 573_8 into base-2.
  3. Convert 3021_4 into base-2.
  4. Convert 528_{9} into base-3.
  5. Convert ABCD_{16} into base-2.

3.1.3 Problem set

  1. Convert 0.67_8 into base-2.

4 Miscellaneous

4.1 Exercises

4.1.1 Problem set

  1. In the binary representation of 2^{99}\times 7+2^{93}\times 9, spell out ninetieth through hundred-fifth digits.
  2. How many natural numbers take 4 digits in their base-7 representations?
  3. A number written in base-2 uses 41 digits. How many digits are required to represent the same number in base-8?
  4. How many whole numbers require 5 digits for base-5 representation and 4 digits for base-9 representation?
  5. The base-7 representation of a number is pqrs_7. If the same number is represented in base-3, what is the smallest number of digits that the representation could have, and what is the biggest number of digits that the representation could have?