Arithmetic with Different Base Number Systems

1. Arithmetic with Different Base Number Systems Mathematical operation such as adding, subtracting, multiplying and dividing can be used regardless of the base number system used. It is important to remember that none of the digits in the solution are greater than or equal to the base. For example with a base three/ ternary number system, the only digits used are 0,1,2. Addition Subtraction Multiplication Division

2. Addition Add 16 and 19 in base 10 (decimal). 16 +19 16 +19 To add, start with adding the 100 column. 1 16 +19 5 6+9=15 Which is equal to 1 X 101 + 5 X 100 Put five ones in ones or 100 column. Put one ten in tens or 101 column. 1 16 +19 5 Add the tens or 101 column. 1+1+1=3 Since this is in the tens column, this means there are 3 tens. Put 3 in ten in tens or 101 column. The answer is 35. 16 +19 35 Next

3. Addition Add 16 and 19 in base 5 (quinary) 31 +34 31 +34 To add, start with adding the 50 column. 1 31 +33 0 1+4=5. Which is equal to 1 X 51 Put a zero in 50 column. Put one in the 51 column. 1 31 +33 0 Add the tens or 101 column. 31 +33 120 1+3+3=7 Which is equal to 1 X 52 and 2 X 51 because it is in the 51 column. Put a two in 51 column. Put one in the 52 column. The answer is 120 in quinary.

4. Subtraction Subtract 25 from 54 in base 10 (decimal). 54 -25 54 -25 To subtract, start with subtracting 100 column. Since 5 is greater than 4, borrow a 10 from the 101 column. There is 14 in the 100 column There is 4 in the 101 column 4 14 54 -25 Subtract each column. 14-5=9 in 100 column 4-2=2 in 101 column 414 54 -25 29 Answer is 29 in decimal. Next

5. Subtraction Subtract 25 from 54 in base 6 (heximal). 130 - 41 130 - 41 To subtract, start with subtracting 60 column. 26 130 - 41 Since 1 is greater than 0, borrow a 6 from the 61 column. There is 6 in the 60 column There is 2 in the 61 column • 26 • 130 • 41 • 5 Subtract 60 column. 6-1=5 in 60 column Then, subtract 61 column. Since 4 is greater than 2, borrow a 6 from the 62 column. There is 8 in 61 column. There is 0 in 62 column • 086 • 130 • 41 • 5 • 086 • 130 • 41 • 45 Subtract 61 column 8-4=4 Answer is 45 in heximal.

6. Multiplication Multiply 19 by 9 in base 10 (decimal). 19 x 9 19 x 9 Start with multiplying 9 x 9 8 19 x 9 1 9 x 9=81 Which is equal to 8 X 101 + 1 X 100 Put 1 in the 100 column Put 8 in the 101 column 8 19 x 9 1 Next multiply 9 x 1 8 19 x 9 171 9 x 1=9 then add 8 to get 17 The answer is 171 in decimal. Next

7. Multiplication Multiply 19 by 9 in base 5 (quinary) 34 x 14 34 x 14 Start with multiplying 4 x 4 3 34 x 14 1 4 x 4=16 Which is equal to 3 X 51 + 1 X 50 Put 1 in the 50 column Put 3 in the 51 column 3 34 x 14 1 Next multiply 3 x 4 3 34 x 14301 3 x 4=12 then add 3 to get 15, which is equal to 3 X 51 Put 0 in the 51 column and 3 in the 52column Next Continued next slide…

8. Multiplication 34 x 14 301 Next multiply 1 x 4 34 x 14 301 40 Since the 1 is in the 51 column put a 0 in 50 column 1 x 4 =4, put in 51 column 34 x 14 301 340 Next multiply 1 x 3 1 x 3=3 put in 52 column Add the two solutions, since it is in base 5 remember that any addition greater or equal to five carries over. The answer is 1141 in quinary. 34 x 14 301 340 1141

9. Division Divide 174 by 6 in base 10 (decimal). Start by seeing how many times 6 goes into 1. Zero times. 6 6 6 6 6 6 174 174 174 174 -12 5 174 -12 54 174 -12 54 54 0 Next see how many times 6 goes into 17 Two times 2 2 29 29 6 x 2= 12 Subtract 12 from 17 which equal 5. Drop the four down. See how many times 6 goes into 54. Nine times. 6 x 9= 54 Subtract 54 from 54, get zero. The answer is 29 . Next

10. Division Divide 174 by 6 in base 7 (septenary). Start by seeing how many times 6 goes into 3. Zero times. 6 6 6 6 6 6 336 -33 06 336 -33 0 336 174 -12 06 6 0 336 336 Next see how many times 6 goes into 33 as base 7 Four times 4 4 41 41 6 x 4= 33 in base 7 Subtract 33 from 33 which equal 0. Drop the six down. See how many times 6 goes into 6. 0ne times. 6 x 1= 6 Subtract 6 from 6, get zero. The answer is 41 in septenary .