# Addition

Start with the units place digit, 5 + 6 = 11 which is 14

_{7}. So, unit digit is 4 and carry over is 1.

_{7}. So, tens digit is 1 and carry over is again 1.

_{7}.

# Subtraction

456_{8}âˆ’ 367

_{8}

Starting with the units digit, since 6 is smaller than 7, we will borrow 1 from the tens place digit. So, now it is 14 (when the base is 10, we get 10 but here base is 8, so we get 8) and

7 subtracted from it = 14 â€“ 7 = 7, which is the units digit.
Next, tens digit is now 4 and we have to subtract 5 from it. We again borrow 1 from hundredâ€™s place digit. So, now it is 12 and 12 â€“ 6 = 6, which is the tens place digit.
Now, hundredâ€™s place digit is 3(4 âˆ’ 1), so 3 âˆ’ 3 = 0

**Note:**

**Another method of doing these kinds of calculations is to convert these values (in whatever base) into decimal system, then do the actual calculation in decimal system itself and finally converting the numbers into the required or given system.**

# Some standard systems of writing

- Decimal system
- Hexa-decimal system
- Octal system
- Binary system

# Comparison of Different Base Systems

It should be understood here that using different bases creates a difference only in the numbers (more than single digit) and not in the digit. This means in that (5)_{10}is same as (5)

_{9}or any other system of writing the numbers.

Example-1

How many different values of

*N*are possible in the following calculation(2)

*Ã— (4)*_{N}*= (8)*_{N}*?*_{N}Solution

Any value of
This concept can be further understood through the mechanism of multiplication. Multiplication is basically the condensed form of addition. So, 2 added up four times will give 8, and hence all the base systems which have the following can be one of the values of

*N*which has 8 as a digit in that system will satisfy this calculation. So, any value of*N*â‰¥ 9 is possible. Hence infinite values of*N*are possible.*N*:0 1 2 3 4 5 6 7 8.

Example-2

How many different values of

*N*are possible in the following calculation (4)*+ (5)*_{N}*= (9)*_{N}*?*_{N}Solution

In this case also, the digits used can be of any system that has the digit 9. So, all the inegral values

*N*â‰¥ 10 are possible.Example-3

How many different values of

*N*are possible in the following calculation. (4)*Ã— (5)*_{N}*= (24)*_{N}*?*_{N}Solution

Obviously, there can be either just one possible value of

*N*or no possible value of*N*.(4)

_{10}Ã— (5)_{10}= (20)_{10}[We know (4)*= (4)*_{N}_{10 }and (5)*= (5)*_{N}_{10}]Now, (24)

*= 2 Ã—*_{N}*N*^{1}+ 4 Ã—*N*^{0}= 20Hence,

*N*= 8. And this will be the only value of*N*.