Binary and quinary: numbers in base 2 and base 5
Put a number on a place-value abacus in base 2 or base 5. See how many beads each rod gets, read the number in the new base, and check it by multiplying back.
Core concept
In base b each place is worth b times the place to its right (1, b, b², b³, …), and every digit is less than b; to convert, divide by b again and again and read the remainders upwards
Topic: Number systems
We count in base 10: the places are worth 1, 10, 100, 1000, and each digit goes from 0 to 9. Other bases work the same way. In base 5 (quinary) the places are worth 1, 5, 25, 125, and the digits are 0 to 4. In base 2 (binary), used inside computers, the places are worth 1, 2, 4, 8, 16, and the only digits are 0 and 1. To write 13 in base 2, divide by 2 again and again: 13 ÷ 2 = 6 remainder 1, 6 ÷ 2 = 3 remainder 0, 3 ÷ 2 = 1 remainder 1, 1 ÷ 2 = 0 remainder 1. Read the remainders from the last one up: 13 = 1101₂. Check: 1×8 + 1×4 + 0×2 + 1×1 = 13.
What you will learn
- Name the place values in base 2 and base 5.
- Convert a base-10 number into base 2 and base 5 by repeated division.
- Convert a base-2 or base-5 number back into base 10.
Before you start
- Know place value in base 10 and division with remainders.
Materials
- Counters or seeds, and a notebook
Equipment
- A paper abacus with rods for the places
Procedure
- Record 13 in base 2.
- Switch to base 5 and record 13 again.
- Record 25 in base 5.
- Record 16 in base 2.
How this simulation is simplified
- Numbers are whole numbers from 1 to 30.