The operations of addition, subtraction, multiplication, and division are defined for counting numbers independent of the system of numeration used to express the numbers. However, the algorithms we use to perform these operations are dependent on the properties of the system of numeration used.
Addition
Adding 6 plus 7 is the mathematical language representation of putting together a collection of six and a collection of seven things and asking how many things are in the total collection. This definition of addition does not depend on the system of numeration used, since 6 and 7 could be written in any system of numeration without changing the question. The answer to thirteen things is also not dependent on the system of numeration. However, as soon as the answer is 6 + 7 = 13, the fact that 6+7 is one group of ten plus three extras uses a base ten system of numeration. Our algorithm for adding two or more digit numbers in base ten is even more dependent on place values. When we “carry a 1 ” we are adding one group of ten to the next higher place.
Example: Add in base 7: 153₇ + 216₇
Solution:
First, write the addition problem in column form. Adding starts with the units column on
the right just as in base ten. Since 3 + 6 = 9 and 9 is one set of 7 plus 2 extras units, the 2
goes in the units column of the answer line and the one set of 7 is “carried over” to the
next column of sets of 7.
₁
1 5 3
+ 2 1 6
__________
9
Next, add the numbers in the seven's column, 1+1+ 5 = 7. Since seven sets of seven is
one set of 49 with nothing left over, carry the 1 to the next column over and put a 0 in the
answer line in the second column.
₁ ₁
1 5 3
+ 2 1 6
__________
0 9
Then add the numbers in the left column, 1+1+2 = 4.
₁ ₁
1 5 3
+ 2 1 6
__________
4 0 9
Since this number is less than 7, there is no need to carry anything to the next column,
and our answer is (409)₇
Example 3: Add in base 16: A28₁₆ + 739₁₆ .
Solution:
First, write the addition problem in column form. Then add the numbers in the rightmost
column. Since 8 + 9 is 17, which is one set of 16 plus 1 left over, write the remainder 1 in
the answer line under the right column and add the 1 set of 16 to the next column.
₁
A 2 8
+ 7 3 9
__________
1
Next, add the second column. Since 1 + 2 + 3 is 6 which is less than 16, simply put the 6
in the answer line for the second column.
₁
A 2 8
+ 7 3 9
__________
6 1
Finally, add A 7 + . From the base 16 chart earlier in this section, we know that A
represents the Hindu-Arabic number 10. Since 10 + 7 = 17 which is one set of 16 plus 1,
put the remainder 1 under the A and 7 column. Place the one set of 16 as the first digit in
the answer.
₁
A 2 8
+ 7 3 9
__________
1 1 6 1
Thus, (A28)₁₆ + (739)₁₆ = (1161)₁₆
Subtraction
The algorithm used for subtraction of multi-digit numbers is the borrowing algorithm that
reverses the “carrying” procedure for addition.
Example 6: Subtract in base 7: 536₇ + 245₇
Solution:
Before starting the subtraction problem, remember that the place values in base 7 are...7² ,7¹ = 7,1. Thus, 536₇ = 5×7² + 3×7¹ + 6. The number is grouped into groups of 7 and
powers of 7. Therefore, if we need to “borrow” from the next higher place, it is group of
7, not 10, that we are borrowing.
To subtract, first write the problem in column form.
5 3 6
- 2 4 5
__________
Begin subtracting with the units place. Since 6 is bigger than 5, we can simply subtract
and record our answer in the answer line.
5 3 6
- 2 4 5
__________
1
Next, try subtracting in the7¹ place. Since 3 is smaller than 4, we cannot subtract without
borrowing from the next larger place. When borrowing from the next larger place, we
have 5×7² = 5×49 things to borrow from, NOT 500! Borrowing one set of 49 from the 5 present, we have 4 sets of 49 left. This is why the 5 is marked out with a 4 above it in
the largest valued place. So, we have borrowed one set of 49 for the 7’s place. One set of
49 is 7 sets of seven, so we add 7 sets of seven to the 3 already present in the 7’s place.
The notation is written as “7+3” instead of “10” to emphasize the fact that we added 7
extra sets.
⁴ ⁷⁺³
5 3 6
- 2 4 5
__________
6 1
Finally, subtract the numbers in the 49’s place. Since 4 is larger than 2, simply subtract
the digits as normal.
5 3 6
- 2 4 5
__________
2 6 1
The final answer is 261₇ .
Multiplication
When we multiply numbers in other bases, we can do it two ways:
First method
Convert both numbers to base 10, then multiply them normally, then convert that back to the desired base. This is usually preferable when multiplying numbers of different bases.
Example:-
multiply 556₆ by 705₉ and express the product in base 4.
- Convert to base 10:
- 4556 = 4 x 6² + 5 x 6¹ + 5 x 6⁰ = 144 + 30 + 5 = 179.
- 7059 = 7 x 9² + 0 x 9¹ + 5 x 9⁰ = 567 + 5 = 572.
- Multiply normally:
- 179 x 572 = 102388.
- Convert to base 4:
- 102388 / 4 = 25597 remainders 0.
- 25597 / 4 = 6399 reminder 1.
- 6399 / 4 = 1599 remainder 3.
- 1599 / 4 = 399 remainders 3.
- 399 / 4 = 99 remainders 3.
- 99 / 4 = 24 remainders 3.
- 24 / 4 = 6 remainders 0.
- 6 / 4 = 1 remainder 2.
- 1 / 4 = 0 remainder 1.
- (reading the remainder backward)
- final answer 120333310₄.
Second method
Build a times table for the base you are using. This only works if the two factors are the same base.
Example
Solve 34₅ x 42₅
Now multiply it as you would with base 10, but using this times table. Remember to use base 5 when carrying as well:
₁
3 4
x 4 2
__________
3
4 x 2 = 13₅ , So write down 3 and carry 1
₁
3 4
x 4 2
__________
1 2 3
3 x 2 = 11₅ , but add the carry of 1 using base 5 addition: 11₅ + 1 = 12₅ write it down (no further digit, so no need to carry)
₃
3 4
x 4 2
__________
1 2 3
1
4 x 4 = 31₅ , So write down 1 and carry 3
₃
3 4
x 4 2
__________
1 2 3
3 0 1
3 x 4 = 22₅ , but add the carry of 3 using base 5 addition: 22₅ + 3₅ = 30₅ write it down (no further digit, so no need to carry)
₃
3 4
x 4 2
____________
1 2 3
+ 3 0 1
____________
3 1 3 3
So, the answer = (3133)₅
Division
The operation of division is the inverse of multiplication. The question 12 3 ? ÷ = asks
what number, when multiplied by 3, gives 12. Therefore, the multiplication table in a
particular base can be used to solve some simple division problems.
Example: Divide 43₈ ÷ 7₈.
Solution:
The multiplication table for base 8 shows that 7₈ × 5₈ = 43₈. Therefore,
43₈ ÷ 7₈ = 5
Example: Divide 11₃ ÷ 2₃.
Solution:
The multiplication table for base 2 shows that 2₃ × 2₃ = 11₃. Therefore,
11₃ ÷ 2₃ = 2₃
Let us take another example with "Logic Kit" app this app is very easy and so good in these subjects of number systems. Download This app from the app store for iOS devices click here.
1- Open the app
2- Choose More Arithmetic
3- Select any base you need
4- write your equation you need to solve
5- You see the solution then click on show steps
Tags:
Number Systems
