**9 + 10 = 21** and **9 + 10 = 19** which is correct? Well, both are correct depending on the number base system the calculation is done. However, if it is in the standard base10 (decimal) system, then 9 + 10 = 19. But what if we modify the equation a bit with other number bases?

Suppose the numbers on the left are in base x and the number on the right is in base y. Then we can say, for what values of x and y is this equation true?

(9 + 10) (base x) = 21 (base y).

## 9 + 10 = 21 proof

First, understand that we have number base systems. Most of our counting is done in the base10 (decimal), which happens to be the most widely used number base system by modern civilizations.

And that is why most people would say, 9+10 = 19. Which is correct in base10 or in other base systems greater than ‘ten’. But when we are counting in a base, less than ‘ten’, it is not correct.

For example, when counting is in base2 or base9; number ‘9’ is not valid and number 10 is not valid. Remember, one zero (10) is different from (10). Infact ‘ten’ supposed to be denoted as ‘A’ because that is the correct symbol in number base system, to avoid confusion.

Similarly, when counting is in base10, number ‘ten’, which is denoted as ‘A’ is not valid, reason we write one zero (10) instead of ‘ten’ which correct symbol is ‘A’.

Example:

**When counting is in base2 (binary**)

1, 10, … (we start a new digit)

**When counting is in base3 (ternary)**

1, 2, 10 (we start a new digit)

**When counting is in base10 (decimal)**

1, 2, …8, 9, 10, (we start a new digit)

**When counting is in base16 (hexadecimal)**

1, 2, 3, …,8, 9, A, B, C, D, E, F, 10 (we start a new digit).

## Convertion of Number base

The good news is, we can convert a number from any base to another.

So, 9 + 10 which is in base10 can be converted to base9.

- When 9 is converted from base10 to base9. We have 9 = 10 (this is one zero, not ten)*
- When ten (10 or A) is converted to from base10 to base9. We have 11.

This means that 9 + 10 in base10 is 10+11 in base9.

Therefore, 10 + 11 = 21 (base9)

**Alternatively,**

9 + 10 = 19

Converting ’19’ from base10 to base 9.

=> 9 + 10 (,base10) = 21 (base9)

Note: It is two one, not twenty one.

Therefore, 9 + 10 = 21. (Q.E.D).

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