Number systems: conversions and games

Writing a number in another base does not change its value: it changes how the digits represent it. The interactive number-systems lab lets you follow each conversion and play with powers of 2.

What does “base” mean?

In the decimal system we use ten digits, from 0 to 9. Each place is worth ten times the place to its right: 347₁₀ = 3×10² + 4×10¹ + 7×10⁰. In base b, places instead have values 1, b, b², b³ and so on. Every digit must be smaller than b. In bases 11 through 16 we use A, B, C, D, E, F for values 10 through 15.

From base x to base 10: add place values

Read the digits from right to left and multiply by successive powers of the base. For example, 2A₁₆ = 2×16¹ + 10×16⁰ = 32 + 10 = 42₁₀. The letter A means 10; it is not a variable.

General formula: (d_n…d₁d₀)_x = d_nxⁿ + … + d₁x + d₀. This calculation also works when some digits are zero.

From base 10 to base y: repeated division

Divide the number repeatedly by y. Write down each remainder, then read them from last to first. Convert 45₁₀ to base 2:

1. 45 ÷ 2 = 22, remainder 1.
2. 22 ÷ 2 = 11, remainder 0.
3. 11 ÷ 2 = 5, remainder 1.
4. 5 ÷ 2 = 2, remainder 1.
5. 2 ÷ 2 = 1, remainder 0.
6. 1 ÷ 2 = 0, remainder 1.

Reading upwards gives 101101₂. Check: 32 + 8 + 4 + 1 = 45.

From base x to base y: base 10 as a bridge

Combine the previous two procedures. For example, 231₄ = 2×4² + 3×4¹ + 1 = 32 + 12 + 1 = 45₁₀. Converting 45 to base 2 by repeated division gives 101101₂. Therefore 231₄ = 45₁₀ = 101101₂.

The six-card number trick

Think of an integer from 1 to 63. We show you six cards, one at a time. For each card, answer only yes or no: is your number on it? Set aside the cards you answered no to. The first numbers on the remaining cards are 1, 2, 4, 8, 16 and 32; add them to recover your number.

Why does it work? These are the six powers of 2, the six places in a binary number. Every number from 1 to 63 has a unique combination of those powers. For example, 37 = 32 + 4 + 1: it appears exactly on the cards starting with 32, 4 and 1. Each card contains 32 numbers, precisely those whose corresponding binary place is 1. Answering no to all cards gives 0, outside the chosen range.

Try the interactive six cards: you can go back to the previous card or start over with another number.

Two more games in the lab

Binary lights: switch eight powers of 2 on or off to make a target from 1 to 255. Base challenge: solve ten conversions among bases 2 to 16, alternating the three methods explained above. After each answer, the correct value appears immediately.

An existing game: the poisoned bottle

Yes/no answers can also be read as binary digits in another problem: each bottle receives a distinct code, and the detector pattern identifies it. To explore this application, try The poisoned bottle, an online game already on the site. Six binary answers distinguish 2⁶ = 64 possibilities; twelve distinguish 2¹² = 4,096.

Try it yourself

1. Convert 10110₂ to base 10.
2. Convert 58₁₀ to base 2.
3. Convert 132₅ to base 16 via base 10.

The lab handles non-negative integers; fractions and negative numbers need further conventions. Remember: another base changes the writing, not the number.