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--- haas:spring2015:cprog:projects:mbe1 [2015/02/06 16:32] – [Background] wedge
+++ haas:spring2015:cprog:projects:mbe1 [2015/02/07 13:31] (current) – wedge
@@ Line 6: / Line 6: @@
 ~~TOC~~
-======Project: MENTAL MATH - MULTIPLY BY 11 (mbe0)======
+======Project: MENTAL MATH - MULTIPLY BY 11 (mbe1)======
 =====Objective=====
@@ Line 142: / Line 142: @@
 Also, the do-while is the only one of our loops which NEEDS a terminating semi-colon (**;**).. please take note of this.
-====Multiplying any single digit number by 11====
-This may be a pattern of which you are already aware- to multiply any single-digit number (base 10) by eleven, you simply duplicate the digit twice.
-In the case of 1 x 11, we get: 11
+====arrays====
+The other important component of our **mbe1** project will involve the effective use of arrays to further aid us in having an efficient (and very importantly: scalable) solution to our problem.
-For 2 x 11, we see: 22
+An array is basically just a collection of variables. Where before we'd declare a separate variable for each number we'd wish to store, an array lets us have but one variable name, but multiple storage location (like PO Boxes-- they're all located at the same "place", the post office, but there are multiple boxes for storing individual values). Addressing/location of information can be greatly simplified through the use of arrays.
-For 3 x 11, we have: 33
+An array is what is known as a **homogeneous** composite data type. It technically is a modifier (or adjective, if you will) of any valid variable data type. We basically use it to say "I'll take X of these", and slap one name on all of them.
-and this trick works all the way through 9 x 11, yielding: 99
+**homogeneous** means all of the same, indicating it can ONLY contain variables of the exact same type (such as only **int**egers, only **short int**egers, only **float**s, only **char**s, etc.)... composite indicates "made up of various parts or elements", or that it is a "container"... it is like a pack of Necco wafers... when you think about it, a pack of Necco wafers ONLY contains necco wafers. You don't see a necco wafer, a pez, an M&M, etc. you ONLY see Necco wafers in a pack of Necco wafers... therefore, a pack of Necco wafers is the same as our array being described (in concept).
-====Multiplying any double digit number by 11====
+An array has a few other requirements:
-Here we do a pivot and then perform simple arithmetic to obtain the middle value.
-In the case of 10 x 11, we take 10 and pivot it, getting 1 and 0, respectively our first and last digit of our soon-to-be solution.
+  * its size, once declared, remains constant (you cannot say you'd like a 4 element array and then later ask to double or halve it--- there is no such thing as "dynamic" arrays or array resizing... even languages that claim to have it, they are lying to you (they are declaring a new array in the background to your altered specifications, copying over all the values, and then presenting that copy as your array)).
+  * arrays are located by their name (just as any variable is), along with an address/index/offset (which mailbox, or position of necco wafer in the pack).
+  * arrays start at an offset of 0.
+  * arrays can be expressed as a pointer (I tend to treat them as pointers), and there's this "bracket notation" we commonly see which is actually trying to hide the pointery truth from you.
+  * C cannot perform global operations on arrays-- you must transact on an array one element at a time (this is the case with all languages, although some will cheat and do the work behind-the-scenes, making you think it is possible, when really they are casting illusions).
-To get the middle value, we add these two values together: 1+0=1
+===Declaring a 4 element integer array===
+Let us see our array in action:
-So, the result of 10 x 11 is: 1 (1+0) 0
+<code c>
-or: 110
+int numbers[4];
+</code>
-Let's try it with 32 x 11:
+This code will declare a new variable, of type **int**, called **numbers**... the brackets indicate we are allocating **four** consecutive **int**-sized units that will be associated with this variable called **numbers**... that is, **numbers** is a post office with 4 mailboxes, addressed 0, 1, 2, 3 (0-3 is 4 distinct values).
-<code>
+To access the first box, we access the **0** offset; the second box is right next to the first, at offset **1** (it is one int away from the first one). Similar with the third and fourth.
-x 11 = 3 (3+2) 2
-        = 3 5     2
+To place a **17** in the first (position **0**) element of our **numbers** array, we'd say the following:
-        = 352
+<code c>
+numbers[0] = 17;
 </code>
-This is almost the entire process, but there's one other factor we need to be aware- if the summing of the first and last values yields a value greater than 10, we must propagate the carry to the next digit to the left (i.e. the first digit).
+To place a **36** in the third (position **2**) element of our **numbers** array, we'd say:
-For example, let us take the maximum two digit value (99):
+<code c>
+numbers[2] = 36;
+</code>
-Using this process as it has been described thus far, we would (incorrectly) get:
+===Using variables as our array index position===
+Because the array index is a number, and things like **int**s are numbers, we can also specify the array location via a variable. To wit, we will assign a 52 to the fourth array element, but do so via an **index** variable we set up:
-<code>
+<code c>
-x 11 = 9 (9 + 9) 9
+int index = 3;
-        = 9 18      9
+numbers[index] = 52;
-        = 9189
 </code>
-But that would be incorrect mathematically.
+Because **index** contains a 3, we're telling the computer we wish to put a 52 in the array element indicated in the **index** variable (the fourth element).
-To compensate (or, to present the full rules for the trick), we take the sum of this result as the middle digit, and apply the carry to the next digit to the left, so:
+===Using variables for array contents===
+As well, because we are putting values in our array elements that conform to particular data types, we can use variables there as well (in this case, put a 96 into the second array element-- using variables to store both the index and the value):
-<code>
+<code c>
-x 11 = 9     (9+9) 9
+int value = 96;
-        = (9+1) 8     9
+int index = 1;
-        = 10    8     9
-        = 1089
+numbers[index] = value;
 </code>
-And we now have the correct result.
+Hopefully these examples have proved useful with respect to basic concepts and syntactic usage of arrays.
-As another example, let us look at 47 x 11:
+We now explore the productive collaboration of arrays and loops:
-<code>
+====Using loops and arrays together for universal harmony====
-x 11 = 4     (4+7) 7
-        = (4+1) 1     7
-        = 5     1     7
-        = 517
-</code>
-Got it? Try it with some other examples.
+To really make the most out of arrays in scaling our algorithms, using them in conjunction with loops gives us the most bang for our buck. The advantage of arrays+loops is that with the ONE consistent variable name, representing many NUMERICALLY-identifiable elements, we can work with ranges of data sets without the need to make tons of exceptions for each possible use case (worst case we just make an array of the maximum expected size, and only use what we need).
-===sum vs. carry===
+===42 everywhere===
-In grade school, when learning to do arithmetic by hand (you still are taught how to do arithmetic by hand, right?), we first learned the concept of sum and carry. This bore value as we were applying this to place values of the number.
-Little did you know then, but you were learning the basics of effective numerical and logical problem solving within the domain of Computer Science in grade school!
+To illustrate, here we will declare an 11 element array (called **data**), and fill each element with the value 42 using a for loop:
-For example, in the case of the number 18, when dissecting the number into its place values, we have:
+<code c>
+int data[11], position = 0;
-  * one 10
+for(position = 0; position < 11; position=position+1) // see, using long form, could have done "position++"
-  * eight 1s
+{
+    data[position] = 42;
+}
+</code>
-In single digit terminology, 18 is expressed as a sum of 8 with a carry of 1. We see this more clearly when producing the value, see the original equation of 9+9:
+===Display array contents===
+What if we wanted to print the contents of the array? Once again, we use a loop, and print out each value, one at a time.
-<code>
+Important considerations:
-   <-- carry (to be added to 10s position)
+  * again, with C, being true to how the computer actually works, we can only access the array one element at a time
+  * because we know array indices start at 0, we have a known starting point
- + 9
+  * because we know how big our array is (11 elements, from previous example), we know how many elements to go for
- ----
+  * each element is located one after the other-- 0 is followed by 1 is followed by 2 etc.
-  <-- sum (of 1s position)
+... therefore, we have all the ingredients for a **for** loop:
+<code c>
+for (position = 0; position < 11; position++)
+{
+    fprintf(stdout, "%d ", data[position]);
+}
+fprintf(stdout, "\n");  // what important role does this line play?
 </code>
-See what is happening here? The basis for adding multiple-digit numbers. Perhaps it would make more sense if we showed how adding 9 + 9 was in fact adding two 2-digit numbers together:
+This should result in the following program output:
-<code>
+<cli>
-   <-- carry (to be added to 10s position)
+42 42 42 42 42 42 42 42 42 42
+</cli>
- +09
- ----
-  <-- sum (of 1s position)
-</code>
+===Backwards?===
-Then we have the follow-up addition to determine the value of the 10s place:
+What if we wanted to display the contents of our array in reverse (from position 10 to position 9, to 8, down to 0)?
-<code>
+We'd still want to use a loop, but look at how we structure it:
+<code c>
- +0
+for (position = 10; position >= 0; position--)
- --
+{
-  <-- sum (of 10s position)
+    fprintf(stdout, "%d ", data[position]);
+}
+fprintf(stdout, "\n");  // what important role does this line play?
 </code>
-and we would technically have a resulting carry of 0 (but adding zero to any values gives us the value itself-- the so-called **additive identity property** we learned in math class).
+Notice how the loop-terminating relational statements differ (comparing the two-- for forward and backward, does it make sense?), and also how we progress between individual elements (in one we are incrementing, in this recent one we are decrementing).
-Once we are all said and done, we concatenate the tens and ones places together:
+That should make sense before you try to proceed.
-(ten) and 8 (ones): 18
+===Thinking with arrays===
+Using arrays in your algorithms represents a potential barrier you have to overcome. Up until this point, we've been getting used to labelling all our variables with unique, individual names.
-====Multiplying any three-digit number by 11====
+Now, with arrays, we have one common name, distinguishable by its element offset. That has been known to cause some conceptual problems due to the mildly abstract nature it creates. It would certainly not hurt to draw some pictures and manually work through some examples, step-by-step... it may be confusing at first, but the more you play with it, ask questions, play, read, etc., the sooner things will start to click.
-In this case we merely extend the pattern from double digits, rippling through a series of comparing each set of two consecutive digits.
-Let's look at 123 x 11:
+As some of you have started to realize with **mbe0**, the actual core work of the project wasn't actually that hard, once you saw through the illusion of complexity we had placed in front of it. By using arrays, we can make our solutions even easier (and code even simpler)... but, we will initially have to eliminate self-imposed mental obstacles making the problem appear significantly more difficult than it actually is.
+====Multiplying a number (of varying digits) by 11====
+In **mbe0**, we specifically looked at 3 usage cases for our mental math problem: 1-, 2-, and 3-digit number. I limited it to those because, lacking arrays and loops for that project, the code would have gotten impossibly long and complex, plus: I wanted you to focus on the basics of variable usage and if-statements.
+Now that we have those down, we can now apply arrays and loops to optimize and enhance a solution, and to allow it to scale to a wider range of possibilities (why limit ourselves to just 1-, 2-, and 3-digit values? Once we see the pattern, we can apply this to 4-, 5-, 6-digit numbers and beyond).
+===3-digits (review)===
+Again, to review, let's look at a 3-digit example. 123 x 11:
 <code>
-x 11 = 1 (1 + 2) (2 + 3) 3
+x 11 = 1       (1 + 2) (2 + 3) 3
-         = 1 3       5       3
+         = (1 + 0) (3 + 0) 5       3  (what are those + 0's? Carry values.)
+         = 1       3       5       3
          = 1353
 </code>
@@ Line 270: / Line 295: @@
 <code>
 x 11 = 5       (5 + 6) (6 + 7) 7
-         = (5 + 1) (1 + 1) 3       7
+         = (5)+1   (11)+1  (13)+0  7  the outside numbers are the carry values
          = 6       2       3       7
          = 6237
@@ Line 284: / Line 309: @@
 A dual benefit of this project is that in addition to extending your programming experience / understanding of C, you could develop this as a mental ability (that is where it originated), and you could then use it as a means of checking your work.
+===4-digits===
+Now let us process a 4-digit example (look for similarities to the 3-digit process, specifically how this is merely an expansion, or an additional step-- due to the additional digit):
+x 11:
+<code>
+x 11 = 4       (4 + 5) (5  + 6) (6 + 7) 7
+          = (4)+1   (9)+1   (11)+1   (13)+0  7   the numbers outside are the carry
+          = 5       0       2        3       7
+          = 50237
+</code>
+Remember, we are processing this from right to left (so that the carry values can properly propagate). While there is no initial carry coming in, we'll add one anyway (0), so we see 13+0 (which is simply 13)... but because we're interested in place values, this is actually a sum of 3, carry of 1... and that one gets sent over to the next place (which has an 11)... so 11+1 will be 12, or sum of 2, carry 1... that carry will propagate to the next position to the left (the 9)... so there's a rippling effect taking place (math in action).
+Can you see how "the same" this process for 4-digit numbers is when comparing to the process for 3-digit numbers? And how the same comparison can be made for 2-digit, and 5-digit, 6-digit, etc.? Please take some time, working through some examples (by hand) to identify and notice the pattern, or essence, of this process. You need to see how it doesn't matter in the long run how many digits- because you're doing the same thing (just a different number of times).
+That "different number of times" will be based on the length of the number... could that be used to help us?
+(Also, the potential exception here would possibly be 1-digit values... if you cannot easily find a way to make 1-digit numbers work with greater-than-1-digit numbers, that's where an if-statement would come into play-- if 1-digit, do this specific process, else do the regular process). I'm not saying one universal solution isn't possible, but at this stage of your structured programming development, such solutions may take a bit more work (and that's okay).
 =====Program=====
-It is your task to write the program that will use the above method to compute the requested one-, two-, or three-digit value against a multiplicand of 11 (without using any multiplication to obtain your result).
+It is your task to write an optimized version of your multiply by eleven program that will use arrays and loops to enable you to enhance and expand the functional capabilities of your program. No longer will you be limited by 1-, 2-, or 3-digit numbers, but you will be able to input up to 8-digit numbers and have your program successfully determine the result (and 8 is merely an arbitrary value I picked, you should easily be able to up it to the tens of thousands and experience no change in functionality) -- actually, our 8-digit limit is considering a data type limitation... the maximum size of an int: **signed int**s can have a maximum value of 2.4 billion, so unless we change to a different data type (or different method of inputting the source number), this will be our limitation.
 Your program should:
   * obtain its input from STDIN.
     * input should be in the form of a single integer value
-  * determine from the input if it is a one-, two-, or three-digit number
+  * determine the number of digits of the inputted value (store this in a variable)
   * perform the correct algorithm against the input
   * propagate any carries
-  * output the final value
+  * use an array (**digit**) to store individual digits from the number input
-    * you can display each digit individually, or combine them into one variable and display that (whichever you prefer)
+  * use another array (**result**) to store the digits of the result number, following manipulations
+    * hint: you will want to make the **result** array one element larger. Why is this?
+  * Display output showing aspects of the process (see example execution below)
+  * output the final value (by iterating through the array, displaying one value at a time)
 =====Execution=====
@@ Line 305: / Line 352: @@
 Enter value: 31415926
 Digits detected: 8
-result[0] = 6
-result[1] =
+Obtaining unique digits, storing in array...
+digit[0] = 6
+digit[1] = 2
+digit[2] = 9
+digit[3] = 5
+digit[4] = 1
+digit[5] = 4
+digit[6] = 1
+digit[7] = 3
+Applying process...
+result[0] = 6 + 0 + 0 (sum of 6, carry out of 0)
+result[1] = 2 + 6 + 0 (sum of 8, carry out of 0)
+result[2] = 9 + 2 + 0 (sum of 1, carry out of 1)
+result[3] = 5 + 9 + 1 (sum of 5, carry out of 1)
+result[4] = 1 + 5 + 1 (sum of 7, carry out of 0)
+result[5] = 4 + 1 + 0 (sum of 5, carry out of 0)
+result[6] = 1 + 4 + 0 (sum of 5, carry out of 0)
+result[7] = 3 + 1 + 0 (sum of 4, carry out of 0)
+result[8] = 3 + 0 + 0 (sum of 3, carry out of 0)
+Displaying result...
 31415926 x 11 = 345575186
 lab46:~/src/cprog/mbe1$
@@ Line 316: / Line 384: @@
 lab46:~/src/cprog/mbe1$ ./mbe1
 Enter value: 7104
+Digits detected: 4
+Obtaining unique digits, storing in array...
+digit[0] = 4
+digit[1] = 0
+digit[2] = 1
+digit[3] = 7
+Applying process...
+result[0] = 4 + 0 + 0 (sum of 4, carry out of 0)
+result[1] = 0 + 4 + 0 (sum of 4, carry out of 0)
+result[2] = 1 + 0 + 0 (sum of 1, carry out of 0)
+result[3] = 7 + 1 + 0 (sum of 8, carry out of 0)
+result[4] = 7 + 0 + 0 (sum of 7, carry out of 0)
+Displaying result...
 x 11 = 78144
 lab46:~/src/cprog/mbe1$
@@ Line 325: / Line 409: @@
 lab46:~/src/cprog/mbe1$ ./mbe1
 Enter value: 56789
+Digits detected: 5
+Obtaining unique digits, storing in array...
+digit[0] = 9
+digit[1] = 8
+digit[2] = 7
+digit[3] = 6
+digit[4] = 5
+Applying process...
+result[0] = 9 + 0 + 0 (sum of 9, carry out of 0)
+result[1] = 8 + 9 + 0 (sum of 7, carry out of 1)
+result[2] = 7 + 8 + 1 (sum of 6, carry out of 1)
+result[3] = 6 + 7 + 1 (sum of 4, carry out of 1)
+result[4] = 5 + 6 + 1 (sum of 2, carry out of 1)
+result[5] = 5 + 1 + 0 (sum of 6, carry out of 0)
+Displaying result...
 x 11 = 624679
 lab46:~/src/cprog/mbe1$
@@ Line 330: / Line 432: @@
 The execution of the program is short and simple- obtain the input, do the processing, produce the output, and then terminate.
-=====Reflection=====
-Be sure to provide any commentary on your opus regarding realizations had and discoveries made during your pursuit of this project.
-  * Does this process work for four digit numbers?
-  * How about five digit numbers?
-  * Do you see a pattern for now this trick could be extended?
 =====Submission=====
@@ Line 341: / Line 437: @@
   * Code must compile cleanly (no warnings or errors)
-  * Output must be correct, and match the form given in the sample output above.
+  * Output must be correct, and resemble the form given in the sample output above.
   * Code must be nicely and consistently indented (you may use the **indent** tool)
   * Code must utilize the algorithm presented above
@@ Line 347: / Line 443: @@
     * have a properly filled-out comment banner at the top
     * have at least 20% of your program consist of **<nowiki>//</nowiki>**-style descriptive comments
-  * Output Formatting (including spacing) of program must conform to the provided output (see above).
   * Track/version the source code in a repository
   * Submit a copy of your source code to me using the **submit** tool.
@@ Line 354: / Line 449: @@
 <cli>
-$ submit cprog mbe1 multby11v2.c mbe1.c
+$ submit cprog mbe1 mbe1.c
 Submitting cprog project "mbe1":
-    -> multby11v2.c(OK)
     -> mbe1.c(OK)

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