VIEWS: 29 PAGES: 18 POSTED ON: 9/12/2012
CALCULATORS: Know your entry system: (1) Infix or Reverse Polish Notation? INFIX: 2 + 3 = 5 (most calculators) RPN: 2 [enter] 3 + 5 (Hewlett-Packard) Infix calculators use parentheses to select the order of operations. “(“ pushes the stack, “)” executes one pending operation. (7-5) * (45) = 1.6 Display S1 S2 S3 ( 7 7 - 7 7 5 5 7 ) 2 - 2 2 ( 2 2 2 4 4 2 2 4 4 2 5 5 4 2 ) 0.8 2 = 1.6 (12 STROKES) RPN calculators do not need parentheses, but have an ENTER key. This indicates that the number is complete and “pushes the stack”. Operations execute immediately. (7-5) * (45) = 1.6 Display S1 S2 S3 7 7 E 7 7 5 5 7 - 2 E 2 2 4 4 2 E 4 4 2 5 5 4 2 0.8 2 x 1.6 (10 STROKES) -Arithmetic or algebraic grouping? ARITHMETIC: 2 + 3 x 4 = 20 (order of entry; requires only one “stack” register; on very simple calculators only) DISP S1 2 2 + 2 2 3 3 2 x 5 5 4 4 5 = 20 A calculator with arithmetic entry usually has NO memory or parentheses. To compute some expressions, such as (2x3) + (5x7) you will need scratch paper. ALGEBRAIC: 2 + 3 x 4 = 14 unary operations then exponents and logs then multiplication and division then addition and subtraction (on all non-RPN scientific calculators) DISP S1 S2 2 2 + 2 2 3 3 2 x 3 3 2 4 4 3 2 = [12 2 this is not shown] 14 A NEW PROBLEM Are higher functions put before or after the numbers they act on? TI83, etc: [ ] 10 , [log] 10, [sin] 45 etc. Most other scientific calculators: 10 [ ], 10 [log], 45 [sin] The second convention is traditional because it requires less memory. However, it is a “reverse notation” like RPN and not really consistent with infix entry. OTHER THINGS TO KNOW: How many levels of parenthesis do you have? None? One? Three? Eight? 20? Do you have memory, and how do you store and recall numbers? Do you have to specify a register? What different angle modes does your calculator have? Degrees/radians/grads? (WARNING: The “mode change” key is NOT the same as the “angle convert” key.) Does your calculator have scientific notation? Engineering notation? How is the exponent indicated? How do you convert? FLOAT: 35462.1 04 SCI: 3.54621 (“ x 104 ”) 03 ENG: 35.4621 always a multiple of 3 WRITTEN ALGORITHMS ADDITION: A simple algorithm: 356 Note that we may end up +869 carrying in the second step 15 as well; this is potentially 11 confusing! 11 1225 The “standard algorithm”: 1 11 “Carrying” (“regrouping”) is 356 indicated with a small digit +869 above the next column to 1225 the left. SUBTRACTION: Standard algorithm: subtract individual digits, regrouping as necessary. Each time you regroup, write 10 (or insert a 1 in the 10’s place) over the digit you’re working on and decrement the one to the left. 4 12 1 14 12 1252 1252 1252 -766 -766 -766 6 86 0 11 14 12 1252 -766 486 Other algorithms: Work left to right. Always subtract the smaller digit from the larger one, but if you take the top one from the bottom write the answer in as a negative. Then total top to bottom. 1252 -766 1000 -500 =500 -10 =490 -4 =486 The middle step may be done mentally (see text, P157) SUBTRACTION: Nines’ complement. Fill out with initial zeros till the numbers are the same length. Subtract each digit of the second number from 9. Add this to the first number. Add 1 and throw away the first digit. 1252 1252 1252 -0766 +9233 +9233 10485 answer: 486 (some people say “move it from the beginning to the end” but this has no real significance. ) CASHIER’S ALGORITHM: Count up from the smaller number to the larger number, adding tokens to a heap at the same rate that you are counting. This is adapted to the particular situation in which (a) tokens are available and (b) the larger number is “round”. $20.00 -$13.57 “fifty eight 1 fifty nine 11 sixty 111 seventy 1 1 1 10 eighty 1 1 1 10 10 ninety 1 1 1 10 10 10 fourteen 1 1 1 10 10 10 10 fifteen 1 1 1 10 10 10 10 $1 twenty” 1 1 1 10 10 10 10 $1 $5 MULTIPLICATION: SIMPLE ALGORITHM: apply the distributive law to all combinations, keeping track of place value (with or without zeros) then add in columns, regrouping if needed. 123 123 x456 x456 18 18 120 12 150 15 600 6 1000 10 Omitting place 1200 12 value zeros may 8000 8 make it easier 5000 5 not to lose place. 40000 4 56088 56088 STANDARD ALGORITHM: Multiply each digit of the second number by the entire first number. (It may be easier to reverse the numbers if the first number has more small digits) 123 x456 738 (=6 x 123) 615 (= 5 x 123) 492 (= 4 x 123) 56088 RUSSIAN PEASANT METHOD: Repeatedly halve one number and double the other. If there is a remainder in halving, ignore it. Add every number in the doubling column that is opposite an odd number. 123 456 456 61 912 912 30 1824 3648 15 3648 7296 7 7296 14592 3 14592 29184 1 29184 56088 This resembles the method used by modern computers! 123 = 64 + 32 + 16 + 8 + 2 + 1 = 11110112 =26 + 25 + 24 + 23 + 21 + 20 123 = 64 + 32 + 16 + 8 + 2 + 1 61 = 32 + 16 + 8 + 4 + 1 30 = 16 + 8 + 4 + 2 15 = 8 + 4 + 2 + 1 7 = 4+2 +1 3 = 2 +1 1 = 1 The repeated-halving algorithm actually computes the base-2 representation of the number. The Russian Peasant Algorithm Continued 123 = 64 + 32 + 16 + 8 + 2 + 1 = 11110112 =26 + 25 + 24 + 23 + 21 + 20 123 x 456 = 20 x 456 +21 x 456 we get these from the repeated doublings +22 x 456 +23 x 456 +24 x 456 +25 x 456 +26 x 456 DIVISION ALGORITHMS: all division algorithms involve some sort of repeated addition or multiplication “till you get there” -simplest: counting up and counting on fingers 20 5: 5,10,15,20: so the answer’s 4 27 5: 5,10,15,20,25,30,oops! 25, answer’s 5, remainder is 2 -more complicated: use bigger blocks: 127 5: 127 20 100 27 5 25 25 2