Learn – Math circle – Base-n arithmetic

Conversion between labels and values

Whole numbers

Exercises

Problem set

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?
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?
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?

Problem set

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

$609$ in base-6
$1847$ in base-7
$2345$ in base-10
$3735$ in base-6
$3470$ in base-4
$11973$ in base-5
$45$ in base-2

Problem set

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

$4512_6$
$4134_5$
$3126_8$
$1232_4$
$10101010_2$

Problem set

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

$12$ in base-8
$12$ in base-7
$17$ in base-9
$20$ in base-6
$14$ in base-7
$18$ in base-8
$21$ in base-5
$24$ in base-6

Problem set

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

$1000_9$
$10000_8$
$100000_5$

Problem set

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

$6^3$ in base-6
$9^5$ in base-9
$8^4$ in base-8
$7^5$ in base-7
$9$ in base-9
$6$ in base-6
$7$ in base-7

Problem set

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

$\left(3\times 6^3 + 5\times 6^2 + 1\times 6 + 4\right)$ in base-6

Problem set

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

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

Fractional numbers

Exercises

Problem set

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

$0.453_6$
$401.234_5$
$215.402_7$

Problem set

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

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

Problem set

In each of the following problems, if $A$ is a digit in the appropriate base, find $A$ .

$0.6 = 0.A_5$
$0.75 = 0.A_8$
$0.1875 = 0.0A_4$
$0.078125 = 0.0A_8$

Problem set

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

$0.875$ in base-6
$0.5536$ in base-5
$0.87890625$ in base-4
$0.591796875$ in base-8
$0.234$ in base-5

Problem set

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

$235.875$ in base-6
$67.8048$ in base-5

Counting and Arithmetic

Exercises

Problem set

Write numbers counting in base-4 up to $1000_4$
Write numbers counting in base-3 up to $1000_3$
Write numbers counting in base-2 up to $100000_2$

Problem set

For each of the following, simplify and give the answer in the same base as is used.

$7_8 + 2_8$
$6_9 + 6_9$
$40_5 + 40_5$
$400_6 + 500_6$
$543_7 + 156_7$
$427_8 + 367_8$
$4423_6 + 1535_6$
$4545_6 + 1011_6$

Problem set

For each of the following, simplify and give the answer in the same base as is used.

$43_8 - 35_8$
$643_7 - 546_7$
$4240_5 - 3423_5$

Problem set

For each of the following, simplify and give the answer in base-8.

$34\times 8^3+12\times8^2+23\times 8$
$42_8\times 1000_8+14_8\times 100_8+27_8\times 10_8$
$42_8\times 8^3+14_8\times8^2+27_8\times 8$

Problem set

For each of the following, simplify and give the answer in the same base as is used.

$1111_2 + 1111_2$
$1111_2 + 1111_2 + 1111_2$
$100001_2 - 1111_2$
$1011_2 \times 111_2$

Problem set

For each of the following, simplify and give the answer in the same base as is used.

$1111_2 \times 1000_2$
$1010101_2 \times 10_2$
$1010101_2 \div 100_2$
$10101010_2 \div 1000_2$

Miscellaneous

Exercises

Problem set

In the binary representation of $2^{99}\times 7+2^{93}\times 9$ , spell out ninetieth through hundred-fifth digits.
How many natural numbers take $4$ digits in their base-7 representations?
A number written in base-2 uses $41$ digits. How many digits are required to represent the same number in base-8?
How many whole numbers require $5$ digits for base-5 representation and $4$ digits for base-9 representation?
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?