5. "Math and Random Numbers" 2025 Winter APS105 Section 3 (University of Toronto)

[Music] e oops alrighty welcome back to APS 105 apparently we're not all supposed to be here all right so we left off last time we could do some multiplication some division some modulo some addition some subtraction but we are Engineers we probably want more math we can do so that's what we get into today and as an Addam added bonus we can start some random numbers so C also comes with the math Library some of you tried to use it in lab one and you were not allowed to because we have not introduced it yet so we will start introducing it today so for more math more advanced than just simple arithmetic we need to use this Library so to add all function prototypes from the library to our code like how we were had to add something to use print F we can do # include math.h on a new line before creating the main function so just to be clear a function prototype tells you the inputs to the function and the output so there's also something come up that may come up called a fun function declaration it's almost the same don't worry about the difference for our purposes if someone says function prototype or function declaration they mean this same thing so like I said a function prototype tells you the input and output types so all function prototypes have the form output type goes first followed by the function name same rules as variable names and then uh parentheses and then all of the input types and again void is a special type that means nothing there can be multiple input types separated by commas and again input types can be void meaning there are no inputs or it could also be void in the output type meaning no output so some examples of function prototypes are well I can write double fu fu is typically just a madeup function name we give just for illustration purposes so I could say FU and then in parentheses I can put a double so this function prototype means means there is a function called Fu that takes a single double as an input and outputs a double this one means there's a function called Fu that takes two doubles as inputs or there's two double arguments if we want to be fancy about it and speak like a c programmer and outputs a double if I write void Fu parentheses double that means there's a function called Fu that takes a single double as an input and outputs notada nothing and then if I have the last variation here if I have double Fu void that means there's a function called Fu that takes no inputs and outputs a double so a function prototype can actually name arguments so you actually know what the arguments are actually doing so for example I could write double Fu double X comma Double Y the names are just to help you understand really at a high level it means Foo still takes two doubles as arguments and outputs a double but now I just gave them names so the first double is called X and the second double is called y they again are just to help you understand which argument is which and the order actually matters because it has to match whenever you use this function so whenever you use the fun function your variable names do not have to match whatever is in the Prototype just the values and the compiler calls these values again literals if I just write a one or a 2.0 or something like that so how the C compiler works so this is kind of breaking apart what the C compiler job is it's not testable but we'll actually understand or help you understand what's actually going on when you compile your program so anything in C that you write with a hashtag it's technically called a pre-processor directive that's just a fancy word of saying hey this is something that happens before c actually compiles your code so there's something called a pre-processor that runs before your code gets compiled and this hashtag tells your pre-processor what to do so it will make some substitution so if I see # include basically all that will do is replace that line with the contents of the specified file so it will do like a copy and paste for you so that std. or sorry stdio.h that's a file somewhere on your computer that spoiler alert has a whole bunch of function prototypes in it like for print F and before you program program gets compiled those get all copied into your program so you don't have to write them and then you can use print F in your program if you delete it there's probably you're probably going to get an error message of the C compiler saying hey I don't know what the hell print f is so header files for the most part contain prototypes of functions and by convention any file that ends in a h like again that stdio.h or PD li. those are called header files that's what the H is supposed to stand for why are they called header files well because they're supposed to be at the top it's the header if I'm writing a letter or something so a library typically comes with one or more header files and the header files contain function prototypes for all usable functions in that Library so we can see what we can use in the math Library so one of the function prototype in the math library that we are allowed to use is well a function to compute the square root of something so in math I'd write you know the square root symbol X in C I have to use a function name I can't just write some squiggly lines and hope it understands it so the function prototype for the square root function is double they had to shorten it of course so they shorten it to sqrt short for square root again C programmers are lazy and they don't like typing very much so this is just what it is and it takes a single double as an argument or as an input and it's called X so this should take the square root of x and the output should be the result of doing the square root operation so if I do square root of four it should give me the answer to what square root of four is which is two so 2 actually use that in a program I could do something like this so again the new thing I have now is this line at the top that has hash include math.h so that brings in the function prototype for square root so I can actually use it and then I can expand on what I did yesterday I can declare a variable called y that is equal to 4.0 so it's a double and then well Square root outputs a double so if I want to actually capture its output I can create a new variable called Yore square root and that will hold I will assign the result of the square root function to that variable so I call square root and I give it as an argument my variable y so by default my VSS code extension will tell me the name that the function prototype used uses here in this kind of gray text so the gray text is just to help you understand which argument is which that's based off the Prototype your names do not have to match this so this is not actually present in my code as I'm typing it it's just what my editor is telling me to just kind of give me a hint as to what argument is which so the X argument to the square root function I'm going to use my value of y for it which is four and then I ass the value of the square root function so the output of the square root function which should be 2.0 to this yqu root function and then if I was to print the result I could again print F I could use the format specifiers for a double and it will do them in order so I could print the square root of the first argument which is y so I will print it using this format specifier which again just only prints off a single digit after the decimal point and then I can say equals again single digit after the decimal point with whatever the result of that function is so like I already ran before but if I just ran it again it will print Square < t of 4 is equal to 2.0 so any questions about that yeah put like a negative number so what if I put a negative number so we can just see what happens if I do the square root of a negative number undefined right or it's imaginary number so let's see what C does so if I just do it I get this special value where it says Nan and you're like what the hell is a nan that's not a number and the answer is exactly Nan stands for not a number so if C can't represent something there's a special double value that means not a number so if you do something invalid it says not a number for anything we do if we ask you to use square root we'll make sure it's positive but yeah things will do odd things depending on it um any other questions yeah a yeah so the question is why did I why didn't I specify that Y is a double so in this case I don't have to specify Y is a double explicitly the function arguments just tell me what the type of Y has to be or the first argument has to be so that X needs to be a double right so in this case if I look at y y is a double so they agree so as long as they agree that's okay so if I said why let's say I declared it as an INT instead see what my compiler does so if I declare it as an INT instead turns out my compiler doesn't complain at me or anything like that if I compile it nothing oh it does complain so part of it complains so it doesn't complain about this but what will happen now is y is an integer and this square root function needs a double as its input right so what c will do is it will implicitly convert that in to a double for us same like when we were doing math so it will take that four and convert it to 4.0 and then it'll do it the begin to thats line a dou yeah so the double at the beginning of line seven that says what the type is of this name right and then since I'm assigning the output of square root to that and it's a double they agree so the output of square root in this case is going to be 2.0 so it assigns it to Yore square root and if they don't agree sometimes c will just do the conversion for you and yell at you in this case it's yelling at me that why is now an INT so Y is now an INT and what my compiler is trying to tell me that it knows that this first format specifier is supposed to be the format specifier for a double but it's telling me that this is an INT so they don't agree anymore so weird things might happen so if I want to make my compiler happy and still use uh y to be an INT I would just remember my format specifier for an INT is just D and then I could do this and make my compiler happy run it and now it's going to say just four instead of 4.0 any other questions with what we are doing yeah sorry so what x stands for is in the prototype in this math.h Library someone defined print or square root something similar to this so they said that square root it's a function that takes a double as an input and they just called it X so that's just the name they give it so that it makes sense to you hopefully and my editor is just showing me this whatever that name is here but it's not actually like I didn't actually write that X there yeah so that X is just my code editor telling me like if I had multiple of them uh we might see it with a different function but if I like had say this took like an X and A Y right I Pro I probably don't want to go back to here to figure out that the first argument is X and the second argument is y I probably want some help from my Editor to tell me which is which all right any other questions all right cool so so just so you have in the slides that's the example that we just went using a square root so if I wanted to and knowing some special things about what's in the math Library just to show you this as an example I could create my own header file in a if I wanted so I could create a header file called square root. with the following content so I could just put inside of it oops I could just put inside of it that square root definition so here in my vs code in that same directory I've Square Ro t.h and here I have that function prototype for just square root so function called square root that takes as an input a double and outputs a double so if I do that create a file called Square root. H well if I want to use it oops so if I want to use it in my C file I can just do hash include and then the convention is or the rule are I use double quotes for anything that is any header files that are in my directory so in this case I just do double quotes and then Square Ro t.h and that will essentially copy paste the contents of that header file into the C file whenever it compiles so let's just get rid of this line because it might be confusing so if I do that and compile nothing changes um it's just square root oops square root custom header still works my compile my program still compiles everything's okay I just don't have math.h there and I just did the function prototype myself if I wanted to again this is not super important for this course you will not be writing header files but that's basically they did so basically how it works is before your code got compiled the contents of this Square Ro t.h was just double Square t double like that so whenever your C pre-processor runs it will include like the contents of those files in this C file before it actually compiles it these files are absolutely huge so I won't show what they do but that including Square Ro t.h is just going to copy and paste basically copy and paste everything from this file which is this one line and replace that include with the contents of the file so it would just do this before my compiler runs and then it would compile this and then C is happy because it knows what the inputs are to the square root function and I can go ahead run my program again like nothing is wrong so just another way to do that a bit above what we need to know for this course but ju again just in case you are interested all right so again like I said before compilation the pre-processor does all the replacements for you and like I said in that case the result after The Replacements are just that square root function declaration gets included there if I gave a name to the input and called it X it would go there so the convention is if that file is in angled brackets that tells the compiler to look for the file in like the default Library directories that exist on your computer and just double quotes file means look for the file in the same directory as the code all right other fun functions we can use is here's one that takes multiple arguments so we can have a function for exponents so in math I would write x to the power of Y in C I need to use a function they didn't call it exponent they called it pow short power so this is a function called pow that takes two arguments they're both a double the first one is called X the second one is called Y and it will do exactly x to the power of Y returning you a double as a result so if I did pow 2.0 comma 8.0 that would be 2 to the^ of 8 which is 256 so questions about that that's just another function with different arguments uh do I have an example well we can just use it here so how my editor would help me is maybe I don't remember which is X and which is y so I could just print off let's just print off whatever the result of some power operation is I could do pow in this case my editor will help me and this case it's not telling me the names oh because I didn't include uh math.h let's go back here so I include math.h then my editor is probably going to help me out a little bit yes indeed so it'll pop up and tell me which argument is which so it will show me the inputs to the function and a little description there return x to the Y power so I could do 2.0 for x and then again we can give a answer your number 10.0 for y and then end it up and then my editor will show me which argument is which again I didn't write X or Y it's just giving me the names to them so at a glance I can tell that I'm doing 2 to the power of 10 so if I compile and run this now I can see that the answer is what I expect 10,24 questions about that cool another function down let's try and make a record so there's trigonomic functions so we love s and Co and all that stuff so here's an example using S so s it's more of a function so this is a function prototype for something called sign takes a a single argument which is a double in this case it's in radians and then it outputs a double also in radians so in math.h if I want to use Pi I probably don't want to remember 3.1415 92 654 I still remember that for some reason I don't know in grade six I decid to remember pi to like 10 digits I don't know I was a weird kid I guess that's why I was in engineering so if you don't want to remember that the C developers help you with that and they have a basically a constant variable for you to use so if I want to use Pi in my code I can use the value mcore Pi all in capitals so technically it's not a variable it's like a pre-processor thing but the compiler will replace or the pre-processor will replace mcore Pi for you with the value of pi and the value of pi in C is as many digits as a double will hold so it's uh up to 21 digits of pi which is more than I know so for example if I wanted to take the S of 90 or whatever that's supposed to be I forget if I wanted to take the S of Pi / 2 I would just do sin M Pi IDE 2.0 and that would just give me the answer of one all right so there's also logarithmic functions to in math if I do the natural log or log base e in C I need to use log annoyingly it's not called Ln and log is not base 10 you just have to remember these things so in C log which takes again one double and outputs a double that is the natural logarithm or to the base e of x if I want to use e like the value of e which is 2 point I forget I haven't used it in a very long time you do capital mcore E and the way to remember that is any constant numbers in the math Library start with M underscores so mathcore Pi mathcore e so if I want to do log base 10 or normal log in C there's a function just called log 10 so log 10 annoyingly log is not log 10 log 10 is log 10 so this will do the common logarithm logarithmic logarithm to the base 10 so of x X so it's just log 10 just remember that that they're slightly different all right so there might be the case where using floating point or using doubles I want to get the remainder still and remember we can only use mod to get a remainder using integers so if I still want that I have to use a function in this case they called it fod which is short for floating mod modulo if you want to remember it that way so this will compute the remainder of x / y or sorry of x / Y and it returns a single number which is just that double so in the case if I want the remainder of 3.5 / 2 I would do fod 3.5 comma 2 and then the result which I just show as an arrow here like I've been doing for the rest of the slides sorry just in case I didn't explain it the remainder is just 1.5 all right other fun ones you can do that might be useful is I can get the minimum and maximum of two values so if I want to know between two values which one is the lower one or which one is the minimum I can use f Min it takes two arguments so in this case they just called them double A and double b and it returns a double so it will return whatever the small smaller value is of the two and on the flip side if I want to see what the larger value is of the two I could use f Max so again it takes two doubles as inputs and outputs a single double what it will output is the larger of the two values there so this probably isn't useful if both arguments are literal values like it's probably silly if I hey C what's the largest number between one or two I should probably be able to figure that out so this might be more useful if the arguments are variables and you don't actually know what the values are so here's some examples so again for this the order of the arguments don't matter so if I do fmin 2.0 comma 3.5 or 3.5 comma 2.0 it doesn't matter it will always return the smaller of the two values so in both of these cases if I use f Min it returns 2.0 if I use f Max again using the same arguments either 2.0 and 3.5 or 3.5 first or 2.0 doesn't matter it's going to return the larger of the two values which in this case is 3.5 yep yeah so the question is can I use more than two values and for this no it's just two values for this one because that's what the Prototype is only has two values there are some special functions like print F that let us have more values but this is not one of them all right other ways all right so getting into rounding some of you got ahead of yourselves in lab one and wanted to round things so there's multiple ways to round to the nearest integer in C and the proper way to do it comes from the math Library so there's a function called round that rounds and computes to the or rounds to the nearest integer and it round rounds away from zero in halfway cases halfway cases are the0 five so rounding away from zero means if they positive numbers it will round up if they are negative numbers it looks like it rounds down so if I round use round for 2.5 it will round up to three if I use round fortive -2.5 going away from zero means it will go to -3 there's also R in which does the more same scientific rounding so that will compute the nearest integer given to you as a double and it rounds towards even in halfway cases so if I round 2.5 it will round to two if I round 3.5 it will round to four and that is by default there's ways to change its rounding Behavior but we're just going to use the default rounding so for the most part for this course our in does more of the proper thing we want it to so here's some examples again rounding to 1.5 again it will always round away from zero so for positive numbers it always rounds up so it Round Up to 2.0 if I do round of 2 point or 2.5 the result will be 3.0 if I use rint for 1.5 it rounds up to 2.0 and if I use R imp for 2.5 which is the halfway case it will round towards even which is 2.0 which is like the proper scientific rounding method so questions about that cool so again like we pointed out before uh the Double format specifier does not change that value so to illustrate that I can just show you what it will do in this case so in this case if I Clare a variable called a and set it equal to 2.5 then the value a is equal to 2.5 if I print f with a format specifier that will just change how I'm outputting that value it won't actually change that value so if I do print F and do the format specifier 0 Z so output no numbers after the decimal place and output a it will actually just print to so it will do the default truncation thing in this case for displaying it won't round or do anything like that so it's just going to Output two and then I didn't actually change the value of a so if I do print f. one so print one digit after the decimal place and that's the rest of the for format specifier double I give it a a still has the value 2.5 so this would actually print 2.5 so that 0 Z the rounding use is technically our in or sometimes it truncates if there's more digits um this 0 Z thing will always round towards zero so it will not try and round it will essentially just truncate so if I want to do some more rounding those rounding functions by default they only just you know they only round to the nearest integer number if I wanted to round to a different decimal point well I could use the printf specifier like you probably did for or what was recommended to do for lab one so again if I want if my value is 2.55 and I do it to one decimal place this will actually print 2.5 and and if I do this only print off a single decimal point and the value is 2.65 it again will do the r in thing and round towards the nearest even value so this will print 2.6 so again the rounding rules only apply to the nearest integer so if I wanted to round that properly or round it differently I have to do a little bit of a trick I have to move the decimal place by just moving it by or multiplying it by 10 so if I multiply a number by 10 that will move the decimal place one place to the right and then I could in this case it would change 2.55 to 25.5 then I can use the round function to turn that into an integer which would round it to maybe 26 if I just used the round function or if I used R int and I always wanted that to round to the proper thing then I just use R int on that value it would properly round it in this case to 26 in both cases and then to get the original value back I just undo multiplying it by 10 so I can divide it by 10 again so the digit I want to round it should be the first digit to the left of the decimal so I just have to move the decimal by multiplying by some power of 10 so moving it by one place multi multiply by 10 if I want to move the decimal by two places I multiply by 100 so after rounding so I do r in it turns it into an integer it is now properly rounded and then I divide by that same number to get the original decimal point so here I am trying to round 2.65 properly to one decimal point so I multiply it by 10 that means I would be rounding 26.5 since our int always uh rounds to the nearest even number the result of this is 26.0 then I divide by 10 to get 2.6 and then now it is properly rounded so print 2.6 and similarly to 2.55 multiply by 10 again like above 25.5 R int would round to 26 and then I divide it by 10 now I finally get it to tell me the answer is properly 2.6 so any questions about that fun so one little trick we have to do so there's some even more fun ways to round to the nearest integer so in math if you've seen the symbol before like the little hat bracket things that's a ceiling in C I can do the same thing but the function is called seal short for sealing and that will always round up to the nearest integer no matter what and in math if I write floor the symbol for floor in C the function is just mercifully called floor they actually wrote it out this time so ceiling will always go up to the nearest uh the next highest integer so ceiling of 2 point or 1.1 goes up to two ceiling of 2.0 it's already in an integer so it'll leave it alone and if it's a negative value it always increases so if it's 1.9 it will go up to the next integer which is just negative uhga 1 similarly for floor floor of 1.9 again it basically does the same thing as truncation so it will just chop it off go down to the nearest uh integer which is 1.0 and for 2.0 again it will stay the same floor does not behave like truncation for uh negative values because it always goes to the lower value so if I do floor of - 1.1 it's kind of in between -2 and -1 the result of that will be -2 so it's slightly different than truncation it's the same for positive numbers though all right so truncation again again similar to floor and ceiling so truncation technically Returns the same as floor for positive values if you want to remember it that way and sealing for negative values but again truncation is what default happens when C converts a float to an integer we just chop off the decimal place if you want to remember like if you want to remember it like that instead so truncation of 1.1 cuts it off it's just one two is just two - 1.5 is just chops off the 0.9 so it's just -1 and again same thing for those values all right so there's lots of other different math functions available you can see the entire list on Wikipedia again this is clickable on my links for this course you should only use the functions that use double for all the arguments and outputs so for some lab possibly lab 2 you might find F abs useful that will give you the absolute value of a number so there's a whole bunch of them that might be important for you so we can use this to solve some problems now so for instance let's solve a problem where we're making change using nickel so given an amount from the user in dollars and cents we want to round to the nearest nickel so for example here is the default sketch of the program I have I'm going to ask the user to input an amount in dollars so my type of that is going to be a double it's going to have a decimal place I'm not going to be concerned about invalid values or the user typing something silly again we're going to be nice to you and only input proper things so I'll do a scan F to actually capture whatever the user inputs into this double variable called amount and then I have a to-do where my job is to round to the nearest nickel so what I like to do when I program is like think about what the user might type and what I should output so some inputs to my program that I might want to think about are like if the user inputs something like 42.4 2 I should think about what my output should be and in this case if I'm rounding to the nearest nickel well my output should probably be in this case 42.4 right that's rounding to the nearest nickel and maybe I want to think about another case where I round up so maybe my input is 42.434652 problem knowing what we know about rounding now anyone think of something we could do to write a program to round our amount to the nearest nickel I'll give you a a minute or two to think about it what we could do to round to the nearest nickel yeah we'll wait 30 seconds and I'll get you okay all right what's your idea okay so your suggestion is to multiply the amount by 200 and then round that and then divide by 200 why 200 okay so so we want to essentially take the in amount we got from the user and then first to break it into two parts maybe multiply it by 100 to get the dollar amount or no to move the decimal so that it would be like you know in this case would be 4242 right so that's the number of cents and then multiply by two or total of 200 to get because you said multiply by 200 right yeah you're getting close to the nearest Fifth so any other ideas so that's kind of there I'm kind of thinking about how many nickel I have yet multiply by multiplying by 20 okay so multiplying by 20 so you want me to multiply it by 20 let's just see so so uh let's just output so we're going to Output a number to two decimal places and you want to take the amount multiply it by 20 and then round that and then divide by 20 something like that all right well the idea is I want to kind of like yeah get groupings of the number of nickels I have and then round that and then shift it back over yeah using the modulo operator the double modulo to get the remainder if I divide it by 0.5 and then round that yeah I could do that well let's see if this one works first so I'll compile it and I called it my program just nickels so enter on amount I'll do 4242 and I got 4240 so seems like it Works 42 43 42 45 that works I would argue 20 in this case does work a way I like to think of about it that might be a bit more obvious multiply it by 20 works or I could do the opposite of that which will essentially do the same thing which is divide by the value of a nickel right if I divide by the value of a nickel basically I'm saying how many nickels is this amount and it will have some decimal point like a decimal of a nickel right and then I round that so that rounds me to the nearest nickel and then instead of dividing it by 20 I do the opposite thing which is I mean it's basically the same thing I'm just switching divide and multiply but I could multiply by the value of a nickel to get the original value back so they mean the same thing I basically just did the reciprocal and did it this to me makes more sense to me because it has a nickel but like if you wrote 20 and just wrote that as a comment and explained your reasoning then that's perfectly good too but they do the same thing if I do this now compile it try it again boom same thing all good we have now written some software that is on any cashier till if you pay cash on the planet so now you can get a job for a cashier company I guess if anyone actually still uses cash which I guess less and less people do now but you've written some useful software now cool all right so any questions about that before we move on to the last topic which is fun cool so you can create random numbers to the standard C library uh defines a function called Rand again it is in this stdlib.h file so its function prototype is Rand takes no inputs and outputs an integer and it outputs a random number between zero and this value that they tell you uh inclusive so this is just some big value that they have defined so if you care about R Rand Max is actually that value using most compilers and this is a Max maximum value represented by an in so it's 2 31 minus 1 so it's just the largest positive value uh represented by an in so always returns a value between zero and that number so it's technically a pseudo random number generator pseudo means not genuine like it's not real so Rand actually uses a formula to determine the next random number and it uses a seed value to create a deterministic set sequence of numbers so if any of you have played games or played Minecraft I guess that's a thing with Minecraft you can set a seed to generate a level for you and the same seed generates the same thing every single time because it's not truly random it's using a formula to come up with a whole bunch of fake random numbers so deterministic means it's the same all the time same input gives you the same output so I can change the seed for Rand using a function called s Rand so s Rand changes that seed value that Rand uses to generate quote unquote random numbers they're not really random so it takes as a single argument something new uh unsigned in seed basically what unsign it it represents a whole number and it doesn't have a sign so it must be negative and it still uses four bytes so an unsigned int can technically represent all it's like there's no sign or anything like that it's basically every bit is used as a number as we did with our numbers column thing so unsigned in can represent any number from zero all the way up to 2 to the 32 minus one so doesn't so there's some numbers in an unsigned int that can't be represented by an INT right anything greater than 2 to the 32 so generally because of this it's a bad idea to Mix signed and unsigned types int by itself is a sign type because it can be negative for this course don't don't ever try to mix them always use an INT if you don't need a decimal don't worry about whether or not it can be negative or not we will tell you whether or not it can be negative don't try and be fancy or anything like that so how I can use this random number generator I can combine it with this modulo operator to generate a different range so modulo using an in gives us the remainder after doing integer division so since Rand returns a whole number the rules with negative values don't apply so I don't have to argue about anything weird so if I just do Rand Mod 20 so Rand will give us some again random number that is between zero and something huge if I do mod 10 it will give us a number between 0 and 9 inclusive because that will be the remainder of doing it and that remainder will essentially be random so this will give us a number between 0 and 9 inclusive so if I want to Generate random numbers between 1 and 10 exclus inclusively all I would have to do is just add a single one to it so if I do Rand mod 10 + one again that will result in a random number between uh 10 or 1 and 10 inclusively so I could use it like that and that would be perfectly fine so here let's see some examples before wrapping up so here's R whoops so here's Rand just generating random numbers we can see that they're all sorts of random giant values that happen to be well in that time they were the exact same because it actually determines the seed from the second time on my clock so if if you run it within the same second it'll generate the same number actually it doesn't actually randomize the seed at all apparently on this operating system so I get the same numbers every single time it's not truly random so I could initialize it to some random seed if I wanted to change the random numbers so again I could use S Rand maybe change it to four and now when I run it I will get a completely different set of random numbers because I changed the seed so now I have a different set of random numbers but every time I run my program they're going to be the same so to summarize this will be a useful formula for you if you can't remember to it so to generalize generating a random number if I want to generate a random number between A and B inclusively where a is greater than B I can use Rand mod a minus b + one that gives me my range and then plus a to give me that starting number and adjust it back so just remember pulling for you we're all in this together