Problem: Are scientific calculators more accurate than other scientific calculators?

General Objective:
o To compare the accuracy of four of the most common calculators used among students by citing their similarities and differences in performing certain computations, and by using an equation with a standard answer.
Specific Objectives:
o To identify four of the most common calculators among students through a survey conducted in SHS.
o To identify some of their similarities and differences, like their input range, etc.
o To find an equation or better yet, try to make our own equation in testing the accuracy of calculators.

Background of the Study
Scientific calculators have contributed much to everyone’s convenience over the years of its discovery. While some people still prefer to solve manually, the rest of the world would rely on calculators, to make their tasks much easier. For a grade school pupil who had just learned his/her multiplication tables, calculators can do his/her job with a guaranteed accuracy. However, when it comes to crucial, lengthy computations which involve trigonometric functions, and other mathematical functions, how dependable could our calculators still be? Shall we have to suffer inaccuracy just because we do not know which calculator can give the most accurate answers to the most complicated equations?
Each of us had an experience wherein in performing some computations which yield large numbers, it was hard to decide on an accurate answer because different calculators yield different results. Like for instance, a calculator would yield an answer of 9864.635493, while another would just yield 9864.6355. It gets confusing as to which calculator has the best answer. Nonetheless, we acknowledge the fact that these calculators have different features. That’s why we would try to work out which of these calculators truly has the most accurate result.
Our group plans to test the accuracy of two of the most commonly used scientific calculators and compare them. We will find accepted equations of testing these calculators wherein there would be an accurate answer. In this process, we are able to enhance our accuracy skills since we will be doing a lot of computations. This is also one way to discover glitches or faults that calculators might have, or might show when performing uncommon operations. Our study could be of use for the math majors or scientists in getting accurate results. Also, it would serve as information for the students so they may be aware that calculators differ in terms of accuracy. We believe that through this study, students will be informed of the limitations of their scientific calculators and learn not to depend on these gadgets all the time because there is no machine that has been invented yet which has 100% efficiency.

Significance of the Study
Today, technology has advanced so far and it will continuously and ceaselessly do so. With the expanding technology, it is already a necessity for calculations to be as accurate as possible. However, calculators tend to reveal incomplete and thus inaccurate answers. In our project, we target to inform people that there are scientific calculators which are less accurate than other scientific calculators. We could also point out the reasons why there are such inaccuracies among calculators.
With our study, the people will be more aware of the different functions and features of calculators. Professionals will be able to distinguish which calculator is ideal for a certain job. Students will be able to identify the most convenient calculator to use for a subject.
In addition to that, people can now distinguish which calculators will give the most accurate computations and which calculators give the least accurate computations. More people will also be aware that inaccurate calculators do exist and what causes them to become “inaccurate”.

Review of Related Literature
History of the Calculator
Chinese abacus
The first calculators were abacuses, and were often constructed as a wooden frame with beads sliding on wires. Abacuses were in use centuries before the adoption of the written Arabic numerals system and are still widely used by merchants and clerks in China and elsewhere.
The 17th century
Wilhelm Schickard built the first automatic calculator called the "Calculating Clock" in 1623. Some 20 years later, in 1643, French philosopher Blaise Pascal invented the calculation device later known as the Pascaline, which was used for taxes in France until 1799. The German philosopher G.W.v. Leibniz also produced a calculating machine.

1930s to 1960s
Mechanical calculator from 1914
From the 1930s through the 1960s, mechanical calculators dominated the desktop computing market These devices were motor-driven, and had movable carriages where results of calculations were displayed by dials.
In 1954, IBM demonstrated a large all-transistor calculator and, in 1957, they released the first commercial all-transistor calculator (the IBM 608). In early 1961, the world's first all-electronic desktop calculator, the Bell Punch/Sumlock Comptometer ANITA (A New Inspiration To Arithmetic) Mk.VII was released.

1970s to mid-1980s
In the early 1970s, the Monroe Epic programmable calculator came on the market. A large desk-top unit, with an attached floor-standing logic tower, it was capable of being programmed to perform many computer-like functions The first hand-held calculator, introduced in January, 1971, was the Sharp EL-8, also marketed as the Facit 1111.
The first pocket calculator with scientific functions that could replace a slide rule was 1972's $395, HP-35 from Hewlett Packard (HP).
The first programmable pocket calculator was the HP-65, in 1974; it had a capacity of 100 instructions, and could store and retrieve programs with a built-in magnetic card reader.

Mid-1980s to present
The first calculator capable of symbolic computation was the HP-28, released in 1987. It was able to, for example, solve quadratic equations symbolically. The first graphing calculator was the Casio fx7000G released in 1985. The CASIO CM-602 Mini Electronic Calculator provided basic functions in the 1970's. The HP 12c financial calculator is still produced. It was introduced in 1981 and is still being made with few changes.
Source: www. wikipedia.com

Materials and Methodology
Procedure 1: Survey
In order to compare the most used calculators in our school we made a survey asking the 3rd and 4th yr. students of School of the Holy Spirit the brand of calculator that they are using. Through this survey, we were able to find out the calculators that most of the students used:

o Casio fx-95MS
o Casio fx-82MS
o Casio fx-82ES
o Sharp EL-531VH

Procedure 2: Derivation of pi
One way of finding out the accuracy of calculators is to compare the calculator’s result with the result of manual calculation.
The materials we used:

For this procedure,
We cut strips of paper with lengths that were the same with circumference of the scotch and masking tapes that we used.
We then labeled the strips of paper and measured them.

After finding out the circumference of the scotch and masking tapes, we measured their diameter with the use of a vernier caliper three times and we got its average.

Then, we manually computed for the pi by dividing the circumference by its diameter. We computed up to the 12th decimal place of the pi.

After manually computing the value of pi, we compared it to the result the calculator gave us.

Procedure 3: Trigonometric Functions and their inverse
Another way of finding the accuracy of a calculator is to key in:
asin(acos(atan(tan(cos(sin(29)))))) and theist value should be equal to 29

Procedure 4: Collection of data
We collected data related to the internal program of the calculator from the official websites of Casio and Sharp.

Observations
I. Comparison of Answers from Procedure I
Calculator Circumference Diameter C/d(manual) C/d(calculator)
Fx 82-MS 126 mm 39.7 mm 3.173803526000 3.173803526000
Sharp EL-531-VH 271 mm 86.06 mm 3.148965837000 3.148965838
292 mm 92.8 mm 3.146551724130 3.146551724
Fx 95-MS 281 mm 89.5 mm 3.139664804469 3.139664804
Fx 82-ES 280 mm 87.07 mm 3.21580337659 3.215803377

II. Comparison of Calculators*
Calculator Internal Digits** Memory***
Fx 82-MS 12 8
Fx 82-ES 15 7
Fx 95-MS 12 9
Sharp EL-531-VH 12 6

*The pieces of information in this table were taken from the manuals in the official websites of the brands of the calculators we used. The said manuals were made available from www.sharpusa.com and http://world.casio.com/calc/download/manual/ .
**Internal Digits- number of digits a calculator is capable of using; also known as Input Range
***Memory- capacity of a calculator to retain values from previous calculations

III. Percentage of Error
Calculator Percentage of Error
Fx 82-MS 1.0253039
Fx 95-MS 0.061365371
Fx 82-ES 1.0253039
Sharp EL-531 VH 0.074705962

IV. Results of Comodore Test
(See values in appendix)

V. Cost of Project
Our project did not cost us anything since all of the materials are either home available or borrowed.

Analysis of Data
Table I:
In this data, we have seen that our Procedure 1 did not prove much. Our calculations are very much the same. Only the last digits differed because the calculators we used are programmed to round-off the tenth digit of the answer. Also, we noticed that our answers were a little different from the accepted pi value of 3.141592654. This discrepancy will be explained in the latter parts of our paper.
This table only showed us that the calculators we are observing round-off to the nearest ninth decimal point.

Table II:
In this table, we found out that Casio Fx 82-MS had the most internal digits. This means that it has the greatest capability to have more accurate answers. The more internal digits a calculator has, the fewer digits the calculator ignored in a computations. The fewer digits overlooked, the smaller amount rounded-off therefore giving a more accurate answer.

Table III:
In the table showing the different percentage errors of the calculators, we saw that there was minimal difference between the results obtained from our manual calculations and those of the calculator’s results. Even though this data did not show us which calculator was the most accurate, it emphasized the fact that our Procedure 1 was not a reliable and accurate experiment. We think that it was such because we realized that errors may have occurred because of human errors and the errors of our measuring tools.

Table IV:
These two tables are representations of the values we acquired from the Comodore test (trigonometric inverses test). There is a repeating pattern seen. From 0 to 89, there is an increase of value. On the 90th degree, the value was undefined, and from the 91st degree to 179th degree, the values decreased. This pattern is also observed on the third and fourth quadrants of the Cartesian plane: the 180th degree returned a value of 0, and the values increased from there to the 269th degree. On the 270th degree, the value was undefined, and from the 271st to the 360th degrees, the values decreased.
In Table IV A, we see that angles 180 degrees apart return the same values. An example of which is the case of the degrees 90 and 270. Both degrees return an undefined value, and they are 180 degrees apart.

Generalizations
A calculator’s accuracy is based on the number of their digits and the variable memory it has. The computation of a calculator is based on the way it is specifically designed to. Errors of calculators are from the way they solve problems (order of operations), the number of internal digits they use and by the memory capacity to be used in long computations. Scientific calculators usually have no problem in computations that require few numbers and operations, but we have to be aware that the result may change if there will be more computation because the errors will add up to one another. Trigonometric functions are good tools to test a calculator's accuracy since they involve many digits.

Recommendations
We, the makers of this study, highly recommend the following to those interested:
• More calculators to be observed
• More ways of proving greater accuracy of a calculator
• Derivation of the equations used in proving the accuracy of calculators

Bibliography
Website Sources
www.sharpusa.com
www.wikipedia.com
http://world.casio.com/calc/download/en/manual

Book Sources
World Book Encyclopedia (2000 Edition)

Values For Comodore Test
Value--Computed Value
1 1.00000001
2 2.00000003
3 3.00000003
4 4.000000011
5 4.999999997
6 6.000000003
7 7.000000014
8 8.000000007
9 9.000000007
10 10.00000001
11 11.00000001
12 12
13 13
14 14
15 15
16 16
17 17.00000001
18 18.00000001
19 19
20 20
21 21
22 22
23 23
24 24
25 25
26 26
27 27
28 28
29 29
30 30
31 31
32 32
33 33
34 34.00000001
35 35
36 36
37 37
38 38
39 39
40 40.00000002
41 41
42 42
43 43
44 44
45 45
46 46
47 47
48 48
49 49
50 50.00000001
51 51
52 52
53 53
54 54
55 55
56 56
57 57
58 58
59 59
60 60
61 61.00000001
62 62
63 63
64 64
65 65
66 66
67 67
68 68
69 69.00000001
70 70
71 71.00000001
72 72
73 73
74 74
75 75
76 76.00000001
77 77.00000001
78 78
79 79
80 80.00000001
81 81.00000001
82 82.00000001
83 83
84 84
85 85.00000003
86 86.00000001
87 87.00000006
88 88.00000007
89 89.00000005
90 Math Error
91 89.00000005
92 88.00000007
93 87.00000006
94 86.00000001
95 85.00000003
96 84
97 83
98 82.00000001
99 81.00000001
100 80.00000001
101 79
102 78
103 77.00000001
104 76.00000001
105 75
106 74
107 73
108 72
109 71.00000001
110 70
111 69.00000001
112 68
113 67
114 66
115 65
116 64
117 63
118 62
119 61.00000001
120 60
121 59
122 58
123 57
124 56
125 55
126 54
127 53
128 52
129 51
130 50.00000001
131 49
132 48
133 47
134 46
135 45
136 44
137 43
138 42
139 41
140 40.00000002
141 39
142 38
143 37
144 36
145 35
146 34.00000001
147 33
148 32
149 31
150 30
151 29
152 28
153 27
154 26
155 25
156 24
156 23
158 22
159 21
160 20
161 19
162 18.00000001
163 17.00000001
164 16
165 15
166 14
167 13
168 12
169 11.00000001
170 10.00000001
171 9.000000007
172 8.000000007
173 7.000000014
174 6.000000003
175 4. 999999997
176 4.000000011
177 3.00000002
178 2.00000003
179 1.00000001
180 0
181 1.00000001
182 2.00000003
183 3.00000003
184 4.000000011
185 4.999999997
186 6.000000003
187 7.000000014
188 8.000000007
189 9.000000007
190 10.00000001
191 11.00000001
192 12
193 13
194 14
195 15
196 16
197 17.00000001
198 18.00000001
199 19
200 20
201 21
202 22
203 23
204 24
205 25
206 26
207 27
208 28
209 29
210 30
211 31
212 32
213 33
214 34.00000001
215 35
216 36
217 37
218 38
219 39
220 40.00000002
221 41
222 42
223 43
224 44
225 45
226 46
227 47
228 48
229 49
230 50.00000001
231 51
232 52
233 53
234 54
235 55
236 56
237 57
238 58
239 59
240 60
241 61.00000001
242 62
243 63
244 64
245 65
246 66
247 67
248 68
249 69.00000001
250 70
251 71.00000001
252 72
253 73
254 74
255 75
256 76.00000001
257 77.00000001
258 78
259 79
260 80.00000001
261 81.00000001
262 82.00000001
263 83
264 84
265 85.00000003
266 86.00000001
267 87.00000006
268 88.00000007
269 89.00000005
270 Math Error
271 89.00000005
272 88.00000007
273 87.00000006
274 86.00000001
275 85.00000003
276 84
277 83
278 82.00000001
279 81.00000001
280 80.00000001
281 79
282 78
283 77.00000001
284 76.00000001
285 75
286 74
287 73
288 72
289 71.00000001
290 70
291 69.00000001
292 68
293 67
294 66
295 65
296 64
297 63
298 62
299 61.00000001
300 60
301 59
302 58
303 57
304 56
305 55
306 54
307 53
308 52
309 51
310 50.00000001
311 49
312 48
313 47
314 46
315 45
316 44
317 43
318 42
319 41
320 40.00000002
321 39
322 38
323 37
324 36
325 35
326 34.00000001
327 33
328 32
329 31
330 30
331 29
332 28
333 27
334 26
335 25
336 24
337 23
338 22
339 21
340 20
341 19
342 18.00000001
343 17.00000001
344 16
345 15
346 14
347 13
348 12
349 11.00000001
350 10.00000001
351 9.000000007
352 8.000000007
353 7.000000014
354 6.000000003
355 4. 999999997
356 4.000000011
357 3.00000002
358 2.00000003
359 1.00000001
360 0

wala lang.haha.halata bang walang magawa?:D