Calculating in Other Bases

1. Calculating in Other Bases MATH 102 Contemporary Math S. Rook

2. Overview • Section 5.3 in the textbook: • Non-base-10 systems • Arithmetic in non-base-10 systems

3. Non-Base-10 Systems

4. Converting Numbers in Non-Base 10 Systems to Base 10 • Consider counting: 1, 2, 3, … in base 10: • What are some of the place names in base 10? • What happens when we reach 9 in the ones place? • What are the possible values for digits in the ones place? • How would we write 5026 in expanded form? • Consider converting a number in a non-base 10 system to base 10 • e.g. What would be the equivalent of 32315 in base 10?

5. Non-Base-10 Systems (Example) Ex 1: Write the base-10 equivalent of the number: a) 11012 b) 21203 c) 23014 d) 3415 e) 5326

6. Converting Numbers in Base 10 to Numbers in Non-Base 10 Systems • To convert a base 10 number such as 172 into a number in a non-base 10 system such as 6: • Divide the base into the number and keep track of the quotientANDremainder • As long as the quotient is not 0, keep dividing the base into the resulting quotient, making sure to keep track of the remainder • When the quotient is equal to 0, write the last remainder • It does not matter if the remainder is equal to 0 • Write the remainders in reverse order • This is the number in the non-base 10

7. Non-Base-10 Systems (Example) Ex 2: Convert the base-10 number 463 into the following bases: a) Base 4 b) Base 5 c) Base 6 d) Base 7

8. Arithmetic in Non-Base-10 Systems

9. Addition in Non-Base-10 Systems • Consider adding 7 + 5 in base 10 • What happens when we exceed the digit 9 when adding in base 10? • When would we have to carry when adding in base 4? • Think of counting in base 4 as 0, 1, 2, 3, (1)0, (1)1, (1)2, (1)3… • e.g. Consider adding 23 + 11 in base 4 • Possible to convert 234 and 114 to base 10, perform the addition, and then reconvert to base 4 • Much easier to do the calculations in base 4

10. Addition in Non-Base-10 Systems (Example) Ex 3: Perform the addition in the given base: a) 34125 + 2315 b) 110112 + 101012

11. Subtraction in Non-Base-10 Systems • Consider subtracting 16 – 7 in base 10 • What happens when we cannot do the subtraction (e.g. 6 – 7)? • What do we do when borrowing in base 10? • How would we borrow in a non-base-10 such as 3? • e.g. Consider subtracting 213 – 123 • Recall that borrowing can propagate across several columns: • Consider subtracting 201 – 79 in base 10 • How would we perform the borrowing? • Consider subtracting 2013 – 123

12. Subtraction in Non-Base-10 Systems (Example) Ex 4: Perform the subtraction in the given base: a) 13256 – 4536 b) 3204 – 2314

13. Multiplication in Non-Base-10 Systems • Consider multiplying 18 x 5 in base 10 • What happens when the product exceeds 10? • How would this apply when multiplying in base 7? • e.g. Consider multiplying 257 x 67 • Consider multiplying by a two-digit number • e.g. Consider multiplying 18 x 15 in base 10 • What happens when we move to the second digit? • e.g. Consider multiplying 257 x 267

14. Multiplication in Non-Base-10 Systems (Example) Ex 5: Perform the multiplication in the given base: a) 415 x 235 b) 3026 x 56

15. Division in Non-Base-10 Systems • Consider dividing 404 / 13 in base 10 • What are the steps for long division? • A good strategy is to create a multiplication table for 13 in base 10 • May take some time at the start, but will save time in the long run • Consider dividing in a non-base-10 system • Definitely construct a multiplication table for the non-base-10 system • e.g. Divide 450036 / 316

16. Division in Non-Base-10 Systems (Example) Ex 6: Perform the division in the given base: a) 34125 / 245 b) 3124 / 34

17. Summary • After studying these slides, you should know how to do the following: • Convert non-base-10 numbers into base 10 • Convert base 10 numbers into non-base-10 numbers • Perform arithmetic in non-base-10 systems • Additional Practice: • See problems in Section 5.3 • Next Lesson: • Exponents & Scientific Notation (Section 6.5)