Addition & Subtraction in Other Bases

Topic Content:

• Addition & Subtraction in Other Bases

11.44_eight – 3.65_eight + 2.61_eight

Apply BODMAS

Rewrite 11.44₈ – 3.65₈ + 2.61₈ as:

⇒ 11.44₈ + 2.61₈ – 3.65₈

Answer = 10.4₈

Work out the base in the following and use it to find the missing figures:

The highest number here is 5, therefore the bases can be 6, 7, etc. …….

Let’s try Base 6

1st Column (from right to left): 2 + 1 = 3. 3 is less than 6 so we write down 3

2nd Column: x + 5 = 3. 3, in this case, is the remainder.
Through trial and error, we know that 4 + 5 = 9 & 9 ÷ 6 = 1 r 3.
So 4 in this case is the missing number and we carry 1 to the 3rd column.

3rd Column: The remainder here is 0. Since we are trying base 6 and 6 ÷ 6 = 1 r 0….. 1 + x + 3 = 6. (remember we are carrying 1 from the 2nd column).
∴ x = 6 – 4 = 2. Carry 1 to the 4th column

4th column: We are carrying 1 from the 3rd column so 1 + 3 + 2 = 6.
Since we are trying base 6, 6 ÷ 6 = 1 r 0.
Write 0 and carry 1 to the 5th column

5th column: 1 + 0 = 1

From our analysis, we can see that the required base is 6 and the missing numbers are 2, 4

∴ 3242₆ + 2351₆ = 10033₆

Class Activity:

Confirm the previous example by solving 10033₆ – 2351₆

Work out the base in the following and use it to find the missing figures:

Explanation

The highest number is 4 so we can try 5, 6, 7 etc

1st Column (from right to left): 1 – 3. We have to borrow.
If we borrow 5 from the 2nd column we get 5 + 1 – 3 = 1, so we know it can’t be 5 because the result is 2, not 4.
Next let’s borrow 6 ∴ 6 + 1 – 3 = 4.
So we can keep trying 6. Remember we borrowed 1 from the 2nd column.

2nd Column: 0 – 1 = -1.
– 1 is less than the result 3 so we have to borrow from the 3rd column∴ 6 + (- 1) – x = 3.
We now have 5 – x = 3. The missing number in this column is 2.

3rd Column = 2 – 1 = 1.
1 is less than the result 4 so we have to borrow from the 4th column ∴ 6 + 1 – x = 4.
We now have 7 – x = 4. The missing number in this column is 3.

4th column = 4 – 1 = 3, ∴ 3 – 3 = 0

We can see that from the equation the numbers are in base 6 and the missing numbers are 3 and 2.

∴ 4201₆ – 3323₆ = 434₆

Find the missing number in the addition of the following numbers, in base seven:

Solution: 1^st step

Solve 4321₇ + 1234₇

The resulting numbers are all less than seven so this addition can be done quickly.

1st Column (from right to left) = 1 + 4 = 5
2nd Column = 2 + 3 = 5
3rd Column = 3 + 2 = 5
4th column = 4 + 1 = 5

Subtract the result obtained from the 1^st step, from the final result to get the missing numbers.

Explanation

1st Column (from right to left): (7 + 1) – 5 = 3
2nd Column: 4 reduces to 3. (7 + 3) – 5 = 5
3rd Column: 3 reduces to 2 = (7 + 2) – 5 = 4
4th column: 2 reduces to 1. (7 + 1) – 5 = 3
5th column: 1 – 1 = 0

∴ Answer = 3453₇