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This version is not peer-reviewed
Research | Research Aim |
[9,10,11] | Explores the application of division by zero in calculus and differentiation |
[12] | Uses classical logic and Boolean algebra to show the problem of division by zero can be solved using today’s mathematics |
[13] | Develops an analogue to Pappus Chain theorem with Division by Zero |
[14] | This paper proposes that the quantum computation being performed by the cancer cell at its most fundamental level is the division by zero. This is the reason for the insane multiplication of cancer cells at its most fundamental scale. |
[15] | Explores evidence to suggest zero does divide zero |
[16] | Considered using division by zero to compare incomparable abstract objects taken from two distinct algebraic spaces |
[17] | Show recent attempts to divide by zero |
[18] | Generalize a problem involving four circles and a triangle and consider some limiting cases of the problem by division by zero. |
[19] | Paper considers computing probabilities from zero divided by itself |
[20,21] | Considers how division by zero is taught on an elementary level |
[22] | Develops a method to avoid division by zero in Newton’s Method |
[23] | This work attempts to solve division by zero using a new form of optimization called Different-level quadratic minimization (DLQM) |
1 | Create an empty stack called “Operator stack” for keeping operators. |
2 | Create an empty list for output. |
3 | Convert the input infix string to a list. |
4 | Scan the token list from left to right. |
5 | If the token is an operand, append it to the end of the output list. |
6 | If the token is a left parenthesis, push it on the “Operator stack”. |
7 | If the token is a right parenthesis, pop the “Operator stack” until the corresponding left parenthesis is removed. Append each operator to the end of the output list. |
8 | If the token is an operator, (), push it on the “Operator stack”. However, first remove any operators already on the “Operator stack” that have higher or equal precedence and append them to the output list. |
9 | When the input expression has been completely processed, check the “Operator stack”. Any operators still on the stack can be removed and appended to the end of the output list. |
1 | User Input: Type of Arithmetic Machine: 1 for “Arithmetic Machine STD” and 2 for “Arithmetic Machine DBZ” |
2 | Create an empty stack called “Operand stack”. |
3 | Convert the string to a list by using the string method split. |
4 | Scan the token list from left to right. |
5 |
|
6 |
|
7 | When the input expression has been completely processed, the result is on the stack. Pop the “Operand stack” and return the value. |
1 | User input: and |
2 | Calculate and |
3 | Calculate and |
4 | If and then |
5 | Else If and then |
6 | Else |
7 | |
8 | If and then |
9 | Else If and then |
10 | Else |
11 | |
12 | |
13 | |
14 | |
15 | |
16 | Return |
Disadvantage | Explanation |
|
Exceptions add minute amounts of time to a program’s run time and can cause the program to run slowly if it occurs frequently enough. The key rule of thumb is that exceptions should occur once every 10,000 calls to a computer program [5]. |
|
As the amount of exception handling increase, it’s harder to know which exceptions are more important than others. This could lead to missing a major issue or dismissing an exception that requires immediate attention [5]. |
|
Inserting exception in the middle of coding makes the code cumbersome and difficult to read especially when there are 100s of lines of code. |
Infix Expression | Postfix Expression |
Semi-structured complex number | Triple format |
1 | User Input: minimum operand value, maximum operand value, equation length L (number of tokens) |
2 | If the equation length is less than 3 then set it to 3 |
3 | If the equation length (L) is even, then add one to it. (That is, set equation length to L+1) |
4 | Create an empty string “Equation” to hold the equation |
5 | |
6 | For T = 1 to Equation Length do the following: |
7 | If T is even then randomly pick an operator from () and add it to the equation string |
8 | Else |
9 | = Randomly pick integer from range (minimum operand value, maximum operand value) |
10 | = Randomly pick integer from range (minimum operand value, maximum operand value) |
11 | = Randomly pick integer from range (minimum operand value, maximum operand value) |
12 | Add to the Equation string |
13 | Return Equation_string |
Equation Length L | Equations in Triple format |
1 | User input: Equation string generated from Equation generator shown inTable 5, Type of Arithmetic Machine: 1 for “Arithmetic Machine STD” and 2 for “Arithmetic Machine DBZ” |
2 | Convert Equation string to Postfix Equation string using Infix to Postfix Algorithm shown in Table 14. |
3 | Evaluate Postfix Equation string using the Postfix Evaluator Algorithm shown in Table 15 and store in Result string. |
4 | |
5 | Return Result string |
Efficiency Aspect | Measure |
Space Complexity |
|
|
|
Time complexity |
|
Characteristics of Equations | Space Complexity | Time Complexity | ||||||||
Simulation No. | Length of Equation | No. operation per Equation | No. of Equations with DBZ operations |
Total Number of DBZ operations | Total Processing Memory (bytes) |
Average Processing Memory Per operation | Total Output memory (bytes) |
Total Processing Time (microseconds) |
No. of operationsper unit time | No. of Equations successfully computed |
1 | 5 | 2 | 25 | 25 | 1628120 | 834.9333 | 2525 | 0.0352 | 55381 | 975 |
2 | 25 | 12 | 36 | 36 | 1626846 | 140.6333 | 3636 | 0.0355 | 326020 | 964 |
3 | 45 | 22 | 77 | 82 | 1761893 | 86.7671 | 7777 | 0.0570 | 356403 | 923 |
4 | 65 | 32 | 86 | 94 | 1874391 | 64.0861 | 8686 | 0.0749 | 390718 | 914 |
5 | 85 | 42 | 134 | 142 | 1989516 | 54.6991 | 13534 | 0.0997 | 364648 | 866 |
6 | 105 | 52 | 140 | 160 | 2130354 | 47.6376 | 14140 | 0.1186 | 377036 | 860 |
7 | 125 | 62 | 176 | 209 | 2256727 | 44.1733 | 17776 | 0.1373 | 372032 | 824 |
8 | 145 | 72 | 189 | 214 | 2388216 | 40.8997 | 19089 | 0.1592 | 366720 | 811 |
9 | 165 | 82 | 215 | 262 | 2549207 | 39.6024 | 21715 | 0.1811 | 355371 | 785 |
10 | 185 | 92 | 237 | 277 | 2775593 | 39.5406 | 23937 | 0.2074 | 338426 | 763 |
11 | 205 | 102 | 276 | 331 | 3042633 | 41.2013 | 27876 | 0.2134 | 346129 | 724 |
12 | 225 | 112 | 280 | 340 | 3211334 | 39.8231 | 28280 | 0.2333 | 345714 | 720 |
13 | 245 | 122 | 311 | 388 | 3563448 | 42.3927 | 31411 | 0.2550 | 329594 | 689 |
14 | 265 | 132 | 314 | 394 | 3791536 | 41.8714 | 31714 | 0.2653 | 341300 | 686 |
15 | 285 | 142 | 327 | 414 | 4055796 | 42.4397 | 33027 | 0.2831 | 337551 | 673 |
16 | 305 | 152 | 335 | 449 | 4393587 | 43.4664 | 33835 | 0.3026 | 333986 | 665 |
17 | 325 | 162 | 363 | 484 | 4569098 | 44.2768 | 36663 | 0.3134 | 329304 | 637 |
18 | 345 | 172 | 357 | 492 | 4996036 | 45.1737 | 36057 | 0.3330 | 332131 | 643 |
19 | 365 | 182 | 419 | 556 | 5405671 | 51.1213 | 42319 | 0.3533 | 299259 | 581 |
20 | 385 | 192 | 437 | 597 | 5936380 | 54.9177 | 44137 | 0.3819 | 283015 | 563 |
Characteristics of Equations | Space Complexity | Time Complexity | ||||||||
Simulation No. | Length of Equation | No. operation per Equation | No. of Equations with DBZ operations |
Total Number of DBZ operations | Total Processing Memory (bytes) |
Average Processing Memory Per operation |
Total Output memory (bytes) |
Total Processing Time (microseconds) |
No. of operations per unit time | No. of Equations successfully computed |
1 | 5 | 2 | 25 | 25 | 1627153 | 813.5765 | 2200 | 0.0373 | 53637 | 1000 |
2 | 25 | 12 | 36 | 36 | 1626846 | 135.5705 | 3168 | 0.0386 | 311235 | 1000 |
3 | 45 | 22 | 77 | 82 | 1761893 | 80.0860 | 6776 | 0.0611 | 360259 | 1000 |
4 | 65 | 32 | 86 | 94 | 1874391 | 58.5747 | 7568 | 0.0801 | 399454 | 1000 |
5 | 85 | 42 | 134 | 142 | 1989516 | 47.3694 | 11792 | 0.1057 | 397431 | 1000 |
6 | 105 | 52 | 140 | 160 | 2130354 | 40.9683 | 12320 | 0.1260 | 412817 | 1000 |
7 | 125 | 62 | 176 | 209 | 2256727 | 36.3988 | 15488 | 0.1461 | 424433 | 1000 |
8 | 145 | 72 | 189 | 214 | 2388216 | 33.1697 | 16632 | 0.1696 | 424473 | 1000 |
9 | 165 | 82 | 215 | 262 | 2549263 | 31.0886 | 18920 | 0.1941 | 422468 | 1000 |
10 | 185 | 92 | 237 | 277 | 2775649 | 30.1701 | 20856 | 0.2212 | 415823 | 1000 |
11 | 205 | 102 | 276 | 331 | 3042633 | 29.8297 | 24288 | 0.2349 | 434266 | 1000 |
12 | 225 | 112 | 280 | 340 | 3211334 | 28.6726 | 24640 | 0.2497 | 448538 | 1000 |
13 | 245 | 122 | 311 | 388 | 3563504 | 29.2090 | 27368 | 0.2824 | 432047 | 1000 |
14 | 265 | 132 | 314 | 394 | 3791536 | 28.7238 | 27632 | 0.2938 | 449330 | 1000 |
15 | 285 | 142 | 327 | 414 | 4055796 | 28.5619 | 28776 | 0.3126 | 454326 | 1000 |
16 | 305 | 152 | 335 | 449 | 4394035 | 28.9081 | 29480 | 0.3368 | 451359 | 1000 |
17 | 325 | 162 | 363 | 484 | 4569322 | 28.2057 | 31944 | 0.3491 | 464055 | 1000 |
18 | 345 | 172 | 357 | 492 | 4996316 | 29.0483 | 31416 | 0.3748 | 458878 | 1000 |
19 | 365 | 182 | 419 | 556 | 5406175 | 29.7043 | 36872 | 0.4053 | 449033 | 1000 |
20 | 385 | 192 | 437 | 597 | 5936828 | 30.9210 | 38456 | 0.4387 | 437607 | 1000 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
Submitted:
03 April 2023
Posted:
03 April 2023
You are already at the latest version
This version is not peer-reviewed
Submitted:
03 April 2023
Posted:
03 April 2023
You are already at the latest version
Research | Research Aim |
[9,10,11] | Explores the application of division by zero in calculus and differentiation |
[12] | Uses classical logic and Boolean algebra to show the problem of division by zero can be solved using today’s mathematics |
[13] | Develops an analogue to Pappus Chain theorem with Division by Zero |
[14] | This paper proposes that the quantum computation being performed by the cancer cell at its most fundamental level is the division by zero. This is the reason for the insane multiplication of cancer cells at its most fundamental scale. |
[15] | Explores evidence to suggest zero does divide zero |
[16] | Considered using division by zero to compare incomparable abstract objects taken from two distinct algebraic spaces |
[17] | Show recent attempts to divide by zero |
[18] | Generalize a problem involving four circles and a triangle and consider some limiting cases of the problem by division by zero. |
[19] | Paper considers computing probabilities from zero divided by itself |
[20,21] | Considers how division by zero is taught on an elementary level |
[22] | Develops a method to avoid division by zero in Newton’s Method |
[23] | This work attempts to solve division by zero using a new form of optimization called Different-level quadratic minimization (DLQM) |
1 | Create an empty stack called “Operator stack” for keeping operators. |
2 | Create an empty list for output. |
3 | Convert the input infix string to a list. |
4 | Scan the token list from left to right. |
5 | If the token is an operand, append it to the end of the output list. |
6 | If the token is a left parenthesis, push it on the “Operator stack”. |
7 | If the token is a right parenthesis, pop the “Operator stack” until the corresponding left parenthesis is removed. Append each operator to the end of the output list. |
8 | If the token is an operator, (), push it on the “Operator stack”. However, first remove any operators already on the “Operator stack” that have higher or equal precedence and append them to the output list. |
9 | When the input expression has been completely processed, check the “Operator stack”. Any operators still on the stack can be removed and appended to the end of the output list. |
1 | User Input: Type of Arithmetic Machine: 1 for “Arithmetic Machine STD” and 2 for “Arithmetic Machine DBZ” |
2 | Create an empty stack called “Operand stack”. |
3 | Convert the string to a list by using the string method split. |
4 | Scan the token list from left to right. |
5 |
|
6 |
|
7 | When the input expression has been completely processed, the result is on the stack. Pop the “Operand stack” and return the value. |
1 | User input: and |
2 | Calculate and |
3 | Calculate and |
4 | If and then |
5 | Else If and then |
6 | Else |
7 | |
8 | If and then |
9 | Else If and then |
10 | Else |
11 | |
12 | |
13 | |
14 | |
15 | |
16 | Return |
Disadvantage | Explanation |
|
Exceptions add minute amounts of time to a program’s run time and can cause the program to run slowly if it occurs frequently enough. The key rule of thumb is that exceptions should occur once every 10,000 calls to a computer program [5]. |
|
As the amount of exception handling increase, it’s harder to know which exceptions are more important than others. This could lead to missing a major issue or dismissing an exception that requires immediate attention [5]. |
|
Inserting exception in the middle of coding makes the code cumbersome and difficult to read especially when there are 100s of lines of code. |
Infix Expression | Postfix Expression |
Semi-structured complex number | Triple format |
1 | User Input: minimum operand value, maximum operand value, equation length L (number of tokens) |
2 | If the equation length is less than 3 then set it to 3 |
3 | If the equation length (L) is even, then add one to it. (That is, set equation length to L+1) |
4 | Create an empty string “Equation” to hold the equation |
5 | |
6 | For T = 1 to Equation Length do the following: |
7 | If T is even then randomly pick an operator from () and add it to the equation string |
8 | Else |
9 | = Randomly pick integer from range (minimum operand value, maximum operand value) |
10 | = Randomly pick integer from range (minimum operand value, maximum operand value) |
11 | = Randomly pick integer from range (minimum operand value, maximum operand value) |
12 | Add to the Equation string |
13 | Return Equation_string |
Equation Length L | Equations in Triple format |
1 | User input: Equation string generated from Equation generator shown inTable 5, Type of Arithmetic Machine: 1 for “Arithmetic Machine STD” and 2 for “Arithmetic Machine DBZ” |
2 | Convert Equation string to Postfix Equation string using Infix to Postfix Algorithm shown in Table 14. |
3 | Evaluate Postfix Equation string using the Postfix Evaluator Algorithm shown in Table 15 and store in Result string. |
4 | |
5 | Return Result string |
Efficiency Aspect | Measure |
Space Complexity |
|
|
|
Time complexity |
|
Characteristics of Equations | Space Complexity | Time Complexity | ||||||||
Simulation No. | Length of Equation | No. operation per Equation | No. of Equations with DBZ operations |
Total Number of DBZ operations | Total Processing Memory (bytes) |
Average Processing Memory Per operation | Total Output memory (bytes) |
Total Processing Time (microseconds) |
No. of operationsper unit time | No. of Equations successfully computed |
1 | 5 | 2 | 25 | 25 | 1628120 | 834.9333 | 2525 | 0.0352 | 55381 | 975 |
2 | 25 | 12 | 36 | 36 | 1626846 | 140.6333 | 3636 | 0.0355 | 326020 | 964 |
3 | 45 | 22 | 77 | 82 | 1761893 | 86.7671 | 7777 | 0.0570 | 356403 | 923 |
4 | 65 | 32 | 86 | 94 | 1874391 | 64.0861 | 8686 | 0.0749 | 390718 | 914 |
5 | 85 | 42 | 134 | 142 | 1989516 | 54.6991 | 13534 | 0.0997 | 364648 | 866 |
6 | 105 | 52 | 140 | 160 | 2130354 | 47.6376 | 14140 | 0.1186 | 377036 | 860 |
7 | 125 | 62 | 176 | 209 | 2256727 | 44.1733 | 17776 | 0.1373 | 372032 | 824 |
8 | 145 | 72 | 189 | 214 | 2388216 | 40.8997 | 19089 | 0.1592 | 366720 | 811 |
9 | 165 | 82 | 215 | 262 | 2549207 | 39.6024 | 21715 | 0.1811 | 355371 | 785 |
10 | 185 | 92 | 237 | 277 | 2775593 | 39.5406 | 23937 | 0.2074 | 338426 | 763 |
11 | 205 | 102 | 276 | 331 | 3042633 | 41.2013 | 27876 | 0.2134 | 346129 | 724 |
12 | 225 | 112 | 280 | 340 | 3211334 | 39.8231 | 28280 | 0.2333 | 345714 | 720 |
13 | 245 | 122 | 311 | 388 | 3563448 | 42.3927 | 31411 | 0.2550 | 329594 | 689 |
14 | 265 | 132 | 314 | 394 | 3791536 | 41.8714 | 31714 | 0.2653 | 341300 | 686 |
15 | 285 | 142 | 327 | 414 | 4055796 | 42.4397 | 33027 | 0.2831 | 337551 | 673 |
16 | 305 | 152 | 335 | 449 | 4393587 | 43.4664 | 33835 | 0.3026 | 333986 | 665 |
17 | 325 | 162 | 363 | 484 | 4569098 | 44.2768 | 36663 | 0.3134 | 329304 | 637 |
18 | 345 | 172 | 357 | 492 | 4996036 | 45.1737 | 36057 | 0.3330 | 332131 | 643 |
19 | 365 | 182 | 419 | 556 | 5405671 | 51.1213 | 42319 | 0.3533 | 299259 | 581 |
20 | 385 | 192 | 437 | 597 | 5936380 | 54.9177 | 44137 | 0.3819 | 283015 | 563 |
Characteristics of Equations | Space Complexity | Time Complexity | ||||||||
Simulation No. | Length of Equation | No. operation per Equation | No. of Equations with DBZ operations |
Total Number of DBZ operations | Total Processing Memory (bytes) |
Average Processing Memory Per operation |
Total Output memory (bytes) |
Total Processing Time (microseconds) |
No. of operations per unit time | No. of Equations successfully computed |
1 | 5 | 2 | 25 | 25 | 1627153 | 813.5765 | 2200 | 0.0373 | 53637 | 1000 |
2 | 25 | 12 | 36 | 36 | 1626846 | 135.5705 | 3168 | 0.0386 | 311235 | 1000 |
3 | 45 | 22 | 77 | 82 | 1761893 | 80.0860 | 6776 | 0.0611 | 360259 | 1000 |
4 | 65 | 32 | 86 | 94 | 1874391 | 58.5747 | 7568 | 0.0801 | 399454 | 1000 |
5 | 85 | 42 | 134 | 142 | 1989516 | 47.3694 | 11792 | 0.1057 | 397431 | 1000 |
6 | 105 | 52 | 140 | 160 | 2130354 | 40.9683 | 12320 | 0.1260 | 412817 | 1000 |
7 | 125 | 62 | 176 | 209 | 2256727 | 36.3988 | 15488 | 0.1461 | 424433 | 1000 |
8 | 145 | 72 | 189 | 214 | 2388216 | 33.1697 | 16632 | 0.1696 | 424473 | 1000 |
9 | 165 | 82 | 215 | 262 | 2549263 | 31.0886 | 18920 | 0.1941 | 422468 | 1000 |
10 | 185 | 92 | 237 | 277 | 2775649 | 30.1701 | 20856 | 0.2212 | 415823 | 1000 |
11 | 205 | 102 | 276 | 331 | 3042633 | 29.8297 | 24288 | 0.2349 | 434266 | 1000 |
12 | 225 | 112 | 280 | 340 | 3211334 | 28.6726 | 24640 | 0.2497 | 448538 | 1000 |
13 | 245 | 122 | 311 | 388 | 3563504 | 29.2090 | 27368 | 0.2824 | 432047 | 1000 |
14 | 265 | 132 | 314 | 394 | 3791536 | 28.7238 | 27632 | 0.2938 | 449330 | 1000 |
15 | 285 | 142 | 327 | 414 | 4055796 | 28.5619 | 28776 | 0.3126 | 454326 | 1000 |
16 | 305 | 152 | 335 | 449 | 4394035 | 28.9081 | 29480 | 0.3368 | 451359 | 1000 |
17 | 325 | 162 | 363 | 484 | 4569322 | 28.2057 | 31944 | 0.3491 | 464055 | 1000 |
18 | 345 | 172 | 357 | 492 | 4996316 | 29.0483 | 31416 | 0.3748 | 458878 | 1000 |
19 | 365 | 182 | 419 | 556 | 5406175 | 29.7043 | 36872 | 0.4053 | 449033 | 1000 |
20 | 385 | 192 | 437 | 597 | 5936828 | 30.9210 | 38456 | 0.4387 | 437607 | 1000 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
Peter Jean Paul
et al.
,
2023
Peter Jean Paul
et al.
,
2024
Daniel Tischhauser
,
2022
© 2024 MDPI (Basel, Switzerland) unless otherwise stated