Arithmetic Algorisms in Different Bases The Addition Algorithm Addition Algorithm - continued To add in any base – Step 1 14 Step 2 1 Arithmetic Algorithms in – Add the digits in the “ones ”column to • Add the digits in the “base” 14 + 32 5 find the number of “1s”. column. + 32 2 + 4 = 6 5 Different Bases • If the number is less than the base place – If the sum is less than the base 1 the number under the right hand column. 6 > 5 enter under the “base” column. • If the number is greater than or equal to – If the sum is greater than or equal 1+1+3 = 5 the base, express the number as a base 6 = 11 to the base express the sum as a Addition, Subtraction, 5 5 = 10 5 numeral. The first digit indicates the base numeral. Carry the first digit Multiplication and Division number of the base in the sum so carry Place 1 under to the “base squared” column and Carry the 1 five to the that digit to the “base” column and write 4 + 2 and carry place the second digit under the 52 column. the second digit under the right hand the 1 five to the “base” column. Place 0 under the column. “fives” column Step 3 “five” column • Repeat this process to the end. The sum is 101 5 Text Chapter 4 – Section 4 No in-class assignment problem In-class Assignment 20 - 1 Example of Addition The Subtraction Algorithm A Subtraction Problem 4 9 352 Find: 1 • 3 + 2 = 5 < 6 →no Subtraction in any base, b 35 2 • 2−4 can not do. 123 123 “6”s to carry. 124 • Start as “1”s column. 7 12 47 • Borrow 1 of the 5 342 6 342 6 – If the subtraction is possible enter the number under “7”s in the minuend 5 0 5 • 2 + 4 = 6 6 = 10 6 the “1”s column. 5 and add 7 to the 2 2 4 9 1 1 One 6 to carry. – If the subtraction is not possible borrow 1 of the then subtract. 123 0 to go in the 6 groups of b in the minuend. 35 2 • 2−4 = 2 column. – The one b being borrowed is worth b in the “1”s 342 • 3−1 = 2 6 column so add b to the digit in the minuend in the “1”s 12 47 505 column . 6 • 1 + 1 + 3 = 5 < 6 → no 62 to carry. • Continue in this manner through the problem 2 2 57 In-class Assignment 20 - 1 No in-class assignment problem In-class Assignment 20 - 2 1 Arithmetic Algorisms in Different Bases Another Subtraction Problem The Multiplication Algorithm Multiplying in Base 3 To Multiply in any base 2011 6 2011 2011 2011 0 1 5 15 • The multiplication algorithm is the same as 212 212 1203 1203 12 03 12 03 212 2123 3 3 in base 10. Things you need to 3 342 342 3 4 2 3 4 2 11022 11022 5 5 5 5 11022 11022 remember. 2011 2011 1 11 2011 3 1 15 – Carry groups of the base 11022 11022 – Indent for each multiplication row 2 x 2 = 4 = 11 3 Remember to 12111023 – Add in the base indent Add in base 3 Carrying in Multiplication The Division Algorithm A Division Problem 1 11 11 2 20 202 2023 4 13202 245 245 245 245 • To divide in any base 4 13202 455 13202 55 55455 13202 13 13 2 2 2 2 – The division algorithm in any base is the 13 13 6 6 6 6 020 020 same as in base 10. Things to remember – 0 02 13 13 4 34 • Digits that are entered in the quotient are at first 2 x 5 =10 =146 5346 22 estimates. 2 1 x 45 = 45 0 x 45 =0 22 2 x 4 =8, 2 x 2 =4, • Do all the multiplications on the side so that the 2 x 45 = 135 Subtract in 8 +1 =9 4 + 1 =5, 0 digits used are not estimates. 3 x 45 = 225 base 5 9 = 1 3 5 = 5 6 6 4 x 45 = 315 3 x 4 =22 • Multiply and subtract in the base used in the 5 5 problem. In-class Assignment 20 - 3 No in-class assignment problem In-class Assignment 20 - 4 2.