Base Calculator

What the base calculator does

This adds, subtracts, multiplies, and divides numbers in any base from 2 to 36, giving the answer in that same base. Pick the base, enter two numbers, choose an operation, and read off the result, all without ever leaving the base you are working in.

It is the most general of the base tools. Where one calculator is fixed to hexadecimal and another to octal, this one lets you set the base to whatever you like, even unusual ones like base 5 or base 20.

How to use it

1. Choose the base. Set the base both numbers are written in, anywhere from 2 to 36.
2. Enter the two numbers and an operation. Type each number in the chosen base and pick add, subtract, multiply, or divide.
3. Press Calculate. The answer appears in the same base.

Press Reset to clear everything.

The same arithmetic in every base

Here is the reassuring truth behind this tool: arithmetic is exactly the same in every base. The column method you learned for ordinary numbers, adding right to left and carrying when a column overflows, works without change in base 5, base 16, or any other. The one and only thing that differs is where a column overflows. In base 10 a column rolls over at 10. In base 8 it rolls over at 8. In base 16 it rolls over at 16. Learn that single adjustment and you can do arithmetic in any base, because everything else stays put.

A worked example with real numbers

Let us add 34 and 13 in base 5, where a column carries as soon as it reaches 5. Work right to left.

• Rightmost column: 4 + 3 = 7. That is 5 or more, so it rolls over: 7 is one 5 and 2 left, giving write 2 and carry 1.
• Next column: 3 + 1, plus the carried 1, is 5. That rolls over exactly: write 0 and carry 1.
• That last carry starts a new column: write 1.

The answer is 102 in base 5. To check it, 34 in base 5 is 19 and 13 in base 5 is 8, and 19 + 8 is 27, while 102 in base 5 is 25 + 0 + 2, which is also 27. The method was ordinary addition; only the carry point moved from 10 down to 5.

Bases that use letters

For bases above 10, there are not enough number symbols, so letters serve as the higher digits: a is 10, b is 11, and so on up to z for 35. You enter them as part of your numbers and the answer comes back using them too. In arithmetic they behave just like the values they stand for, so adding a and 1 in a high enough base gives b, since 10 plus 1 is 11.

The hex and octal calculators

If you only ever work in one base, there are focused versions of this tool. The hex calculator is this calculator fixed to base 16, and the octal calculator is the same fixed to base 8, each tuned to its own setting so you do not have to choose the base each time. This general calculator is the one to reach for when your base is something other than those two, or when you want the freedom to switch bases as you go.

Questions people ask

What can it calculate?

Addition, subtraction, multiplication, and division, on two numbers in any base from 2 to 36, with the answer given in that same base.

Does arithmetic change from base to base?

No. The method is identical everywhere. The only thing that changes is the point at which a column carries, which is the base itself: 10 in decimal, 8 in octal, 16 in hex.

Why do some numbers use letters?

Bases above 10 need digits beyond 9, so letters fill the gap: a is 10 up to z for 35. They act as those values in the arithmetic.

How is this different from the hex and octal calculators?

Those are this same calculator locked to one base, 16 for hex and 8 for octal. This one lets you set any base, which is what you want for other bases or for switching between them.

References

1. Positional notation. Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Positional_notation