CALCULATORS:
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) * (4(5) = 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) * (4(5) = 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
SCI: 3.54621 04 (“ x 104 ”)
ENG: 35.4621 03 always a
multiple of 3
WRITTEN ALGORITHMS
ADDITION:
A simple algorithm:
356
+869
15
11
11
1225
The “standard algorithm”:
1 1 1
356
+869
1225
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.
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 1 1
sixty 1 1 1
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
x456
18
120
150
600
1000
1200
8000
5000
40000
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
+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
-----------------------
Note that we may end up carrying in the second step as well; this is potentially confusing!
“Carrying” (“regrouping”) is indicated with a small digit above the next column to the left.
1252
-766
4 12
1252
-766
6
1 14 12
1252
-766
86
0 11 14 12
1252
-766
486
123
x456
18
12
15
6
10
12
8
5
4
56088
Omitting place value zeros may make it easier not to lose place.
we get these from the repeated doublings
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