void main()
{
int a;
char *x;
x = (char*)&a;
a = 500;
x[0] = 2;
x[1] = 2;
x[2] = 0;
cout<<"final op : "<<a;
}
I know the answer is 514 . But how does it work ?
First, note that exactly how this works is not dictated by the C or C++ standards, and you really shouldn't rely on any specific behavior, unless you really know what you're doing and the ramifications thereof. (since you had to ask, you don't fall into that category, even after reading this answer!)
The two most common ways this is done are both that the integer is stored in base 256, storing one digit per byte (bytes are usually 8-bits, and 2^8 = 256). So when you access it via char*, each place represents one digit.
In little endian format, you access the digits in order of magnitude; e.g. x[0] has the one's place, x[1] has the 256's place, x[2] has the 256^2's place, and so forth. That is, you access the digits from right to left.
In big endian format, the digits are in the opposite order.
Which format is used usually dictated by the computer hardware.
How many base-256 digits make up an int can also vary. 4 digits is common, but 2 and 8 happen, and there are even some exotic environments where int doesn't fit exactly into this picture at all.
On your particular computer and programming environment, what probably happened was that int was at least 4 digits and stored in little endian form; so assigning 500 means the digits were
0, 0, 1, 244
(because 500 = 244 * 1 + 1 * 256; remember how to compute change of base!) and then the assignments changed the digits to
0, 0, 2, 2
which represents 2 * 1 + 2 * 256 = 514.
(also, note that if char is signed in your programming environment, you would probably have read the rightmost digit of 500 as being -12 rather than 244: look up two's complement notation)
The way this works is that x points to the same location that holds the data for a. When you write to or read from x[0], you are accessing the first byte of the value held in a.
This isn't converting between char array and int, this is "accessing an int through a char pointer".
Related
I am currently working on an arbitrary size integer library for learning purposes.
Each number is represented as uint32_t *number_segments.
I have functional arithmetic operations, and the ability to print the raw bits of my number.
However, I have struggled to find any information on how I could convert my arbitrarily long array of uint32 into the correct, and also arbitrarily long base 10 representation as a string.
Essentially I need a function along the lines of:
std::string uint32_array_to_string(uint32_t *n, size_t n_length);
Any pointers in the right direction would be greatly appreciated, thank you.
You do it the same way as you do with a single uint64_t except on a larger scale (bringing this into modern c++ is left for the reader):
char * to_str(uint64_t x) {
static char buf[23] = {0}; // leave space for a minus sign added by the caller
char *p = &buf[22];
do {
*--p = '0' + (x % 10);
x /= 10;
} while(x > 0);
return p;
}
The function fills a buffer from the end with the lowest digits and divides the number by 10 in each step and then returns a pointer to the first digit.
Now with big nums you can't use a static buffer but have to adjust the buffer size to the size of your number. You probably want to return a std::string and creating the number in reverse and then copying it into a result string is the way to go. You also have to deal with negative numbers.
Since a long division of a big number is expensive you probably don't want to divide by 10 in the loop. Rather divide by 1'000'000'000 and convert the remainder into 9 digits. This should be the largest power of 10 you can do long division by a single integer, not bigum / bignum. Might be you can only do 10'000 if you don't use uint64_t in the division.
I'm new to C++ and is trying to learn the concept of array. I saw this code snippet online. For the sample code below, does it make any difference to declare:
unsigned scores[11] = {};
unsigned grade;
as:
int scores[11] = {};
int grade;
I guess there must be a reason why score[11] = {}; and grade is declared as unsigned, but what is the reason behind it?
int main() {
unsigned scores[11] = {};
unsigned grade;
while (cin >> grade) {
if (0 <= grade <= 100) {
++scores[grade / 10];
}
}
for (int i = 0; i < 11; i++) {
cout << scores[i] << endl;
}
}
unsigned means that the variable will not hold a negative values (or even more accurate - It will not care about the sign-). It seems obvious that scores and grades are signless values (no one scores -25). So, it is natural to use unsigned.
But note that: if (0 <= grade <= 100) is redundant. if (grade <= 100) is enough since no negative values are allowed.
As Blastfurnace commented, if (0 <= grade <= 100) is not right even. if you want it like this you should write it as:
if (0 <= grade && grade <= 100)
Unsigned variables
Declaring a variable as unsigned int instead of int has 2 consequences:
It can't be negative. It provides you a guarantee that it never will be and therefore you don't need to check for it and handle special cases when writing code that only works with positive integers
As you have a limited size, it allows you to represent bigger numbers. On 32 bits, the biggest unsigned int is 4294967295 (2^32-1) whereas the biggest int is 2147483647 (2^31-1)
One consequence of using unsigned int is that arithmetic will be done in the set of unsigned int. So 9 - 10 = 4294967295 instead of -1 as no negative number can be encoded on unsigned int type. You will also have issues if you compare them to negative int.
More info on how negative integer are encoded.
Array initialization
For the array definition, if you just write:
unsigned int scores[11];
Then you have 11 uninitialized unsigned int that have potentially values different than 0.
If you write:
unsigned int scores[11] = {};
Then all int are initialized with their default value that is 0.
Note that if you write:
unsigned int scores[11] = { 1, 2 };
You will have the first int intialized to 1, the second to 2 and all the others to 0.
You can easily play a little bit with all these syntax to gain a better understanding of it.
Comparison
About the code:
if(0 <= grade <= 100)
as stated in the comments, this does not do what you expect. In fact, this will always evaluate to true and therefore execute the code in the if. Which means if you enter a grade of, say, 20000, you should have a core dump. The reason is that this:
0 <= grade <= 100
is equivalent to:
(0 <= grade) <= 100
And the first part is either true (implicitly converted to 1) or false (implicitly converted to 0). As both values are lower than 100, the second comparison is always true.
unsigned integers have some strange properties and you should avoid them unless you have a good reason. Gaining 1 extra bit of positive size, or expressing a constraint that a value may not be negative, are not good reasons.
unsigned integers implement arithmetic modulo UINT_MAX+1. By contrast, operations on signed integers represent the natural arithmetic that we are familiar with from school.
Overflow semantics
unsigned has well defined overflow; signed does not:
unsigned u = UINT_MAX;
u++; // u becomes 0
int i = INT_MAX;
i++; // undefined behaviour
This has the consequence that signed integer overflow can be caught during testing, while an unsigned overflow may silently do the wrong thing. So use unsigned only if you are sure you want to legalize overflow.
If you have a constraint that a value may not be negative, then you need a way to detect and reject negative values; int is perfect for this. An unsigned will accept a negative value and silently overflow it into a positive value.
Bit shift semantics
Bit shift of unsigned by an amount not greater than the number of bits in the data type is always well defined. Until C++20, bit shift of signed was undefined if it would cause a 1 in the sign bit to be shifted left, or implementation-defined if it would cause a 1 in the sign bit to be shifted right. Since C++20, signed right shift always preserves the sign, but signed left shift does not. So use unsigned for some kinds of bit twiddling operations.
Mixed sign operations
The built-in arithmetic operations always operate on operands of the same type. If they are supplied operands of different types, the "usual arithmetic conversions" coerce them into the same type, sometimes with surprising results:
unsigned u = 42;
std::cout << (u * -1); // 4294967254
std::cout << std::boolalpha << (u >= -1); // false
What's the difference?
Subtracting an unsigned from another unsigned yields an unsigned result, which means that the difference between 2 and 1 is 4294967295.
Double the max value
int uses one bit to represent the sign of the value. unsigned uses this bit as just another numerical bit. So typically, int has 31 numerical bits and unsigned has 32. This extra bit is often cited as a reason to use unsigned. But if 31 bits are insufficient for a particular purpose, then most likely 32 bits will also be insufficient, and you should be considering 64 bits or more.
Function overloading
The implicit conversion from int to unsigned has the same rank as the conversion from int to double, so the following example is ill formed:
void f(unsigned);
void f(double);
f(42); // error: ambiguous call to overloaded function
Interoperability
Many APIs (including the standard library) use unsigned types, often for misguided reasons. It is sensible to use unsigned to avoid mixed-sign operations when interacting with these APIs.
Appendix
The quoted snippet includes the expression 0 <= grade <= 100. This will first evaluate 0 <= grade, which is always true, because grade can't be negative. Then it will evaluate true <= 100, which is always true, because true is converted to the integer 1, and 1 <= 100 is true.
Yes it does make a difference. In the first case you declare an array of 11 elements a variable of type "unsigned int". In the second case you declare them as ints.
When the int is on 32 bits you can have values from the following ranges
–2,147,483,648 to 2,147,483,647 for plain int
0 to 4,294,967,295 for unsigned int
You normally declare something unsigned when you don't need negative numbers and you need that extra range given by unsigned. In your case I assume that that by declaring the variables unsigned, the developer doesn't accept negative scores and grades. You basically do a statistic of how many grades between 0 and 10 were introduced at the command line. So it looks like something to simulate a school grading system, therefore you don't have negative grades. But this is my opinion after reading the code.
Take a look at this post which explains what unsigned is:
what is the unsigned datatype?
As the name suggests, signed integers can be negative and unsigned cannot be. If we represent an integer with N bits then for unsigned the minimum value is 0 and the maximum value is 2^(N-1). If it is a signed integer of N bits then it can take the values from -2^(N-2) to 2^(N-2)-1. This is because we need 1-bit to represent the sign +/-
Ex: signed 3-bit integer (yes there are such things)
000 = 0
001 = 1
010 = 2
011 = 3
100 = -4
101 = -3
110 = -2
111 = -1
But, for unsigned it just represents the values [0,7]. The most significant bit (MSB) in the example signifies a negative value. That is, all values where the MSB is set are negative. Hence the apparent loss of a bit in its absolute values.
It also behaves as one might expect. If you increment -1 (111) we get (1 000) but since we don't have a fourth bit it simply "falls off the end" and we are left with 000.
The same applies to subtracting 1 from 0. First take the two's complement
111 = twos_complement(001)
and add it to 000 which yields 111 = -1 (from the table) which is what one might expect. What happens when you increment 011(=3) yielding 100(=-4) is perhaps not what one might expect and is at odds with our normal expectations. These overflows are troublesome with fixed point arithmetic and have to be dealt with.
One other thing worth pointing out is the a signed integer can take one negative value more than it can positive which has a consequence for rounding (when using integer to represent fixed point numbers for example) but am sure that's better covered in the DSP or signal processing forums.
I have instructions on creating a checksum of a message described like this:
The checksum consists of a single byte equal to the two’s complement sum of all bytes starting from the “message type” word up to the end of the message block (excluding the transmitted checksum). Carry from the most significant bit is ignored.
Another description I found was:
The checksum value contains the twos complement of the modulo 256 sum of the other words in the data message (i.e., message type, message length, and data words). The receiving equipment may calculate the modulo 256 sum of the received words and add this sum to the received checksum word. A result of zero generally indicates that the message was correctly received.
I understand this to mean that I sum the value of all bytes in message (excl checksum), get modulo 256 of this number. get twos complement of this number and that is my checksum.
But I am having trouble with an example message example (from design doc so I must assume it has been encoded correctly).
unsigned char arr[] = {0x80,0x15,0x1,0x8,0x30,0x33,0x31,0x35,0x31,0x30,0x33,0x30,0x2,0x8,0x30,0x33,0x35,0x31,0x2d,0x33,0x32,0x31,0x30,0xe};
So the last byte, 0xE, is the checksum. My code to calculate the checksum is as follows:
bool isMsgValid(unsigned char arr[], int len) {
int sum = 0;
for(int i = 0; i < (len-1); ++i) {
sum += arr[i];
}
//modulo 256 sum
sum %= 256;
char ch = sum;
//twos complement
unsigned char twoscompl = ~ch + 1;
return arr[len-1] == twoscompl;
}
int main(int argc, char* argv[])
{
unsigned char arr[] = {0x80,0x15,0x1,0x8,0x30,0x33,0x31,0x35,0x31,0x30,0x33,0x30,0x2,0x8,0x30,0x33,0x35,0x31,0x2d,0x33,0x32,0x31,0x30,0xe};
int arrsize = sizeof(arr) / sizeof(arr[0]);
bool ret = isMsgValid(arr, arrsize);
return 0;
}
The spec is here:= http://www.sinet.bt.com/227v3p5.pdf
I assume I have misunderstood the algorithm required. Any idea how to create this checksum?
Flippin spec writer made a mistake in their data example. Just spotted this then came back on here and found others spotted too. Sorry if I wasted your time. I will study responses because it looks like some useful comments for improving my code.
You miscopied the example message from the pdf you linked. The second parameter length is 9 bytes, but you used 0x08 in your code.
The document incorrectly states "8 bytes" in the third column when there are really 9 bytes in the parameter. The second column correctly states "00001001".
In other words, your test message should be:
{0x80,0x15,0x1,0x8,0x30,0x33,0x31,0x35,0x31,0x30,0x33,0x30, // param1
0x2,0x9,0x30,0x33,0x35,0x31,0x2d,0x33,0x32,0x31,0x30,0xe} // param2
^^^
With the correct message array, ret == true when I try your program.
Agree with the comment: looks like the checksum is wrong. Where in the .PDF is this data?
Some general tips:
Use an unsigned type as the accumulator; that gives you well-defined behavior on overflow, and you'll need that for longer messages. Similarly, if you store the result in a char variable, make it unsigned char.
But you don't need to store it; just do the math with an unsigned type, complement the result, add 1, and mask off the high bits so that you get an 8-bit result.
Also, there's a trick here, if you're on hardware that uses twos-complement arithmetic: just add all of the values, including the checksum, then mask off the high bits; the result will be 0 if the input was correct.
The receiving equipment may calculate the modulo 256 sum of the received words and add this sum to the received checksum word.
It's far easier to use this condition to understand the checksum:
{byte 0} + {byte 1} + ... + {last byte} + {checksum} = 0 mod 256
{checksum} = -( {byte 0} + {byte 1} + ... + {last byte} ) mod 256
As the others have said, you really should use unsigned types when working with individual bits. This is also true when doing modular arithmetic. If you use signed types, you leave yourself open to a rather large number of sign-related mistakes. OTOH, pretty much the only mistake you open yourself up to using unsigned numbers is things like forgetting 2u-3u is a positive number.
(do be careful about mixing signed and unsigned numbers together: there are a lot of subtleties involved in that too)
I am learning C/C++ programming & have encountered the usage of 'Bit arrays' or 'Bit Vectors'. Am not able to understand their purpose? here are my doubts -
Are they used as boolean flags?
Can one use int arrays instead? (more memory of course, but..)
What's this concept of Bit-Masking?
If bit-masking is simple bit operations to get an appropriate flag, how do one program for them? is it not difficult to do this operation in head to see what the flag would be, as apposed to decimal numbers?
I am looking for applications, so that I can understand better. for Eg -
Q. You are given a file containing integers in the range (1 to 1 million). There are some duplicates and hence some numbers are missing. Find the fastest way of finding missing
numbers?
For the above question, I have read solutions telling me to use bit arrays. How would one store each integer in a bit?
I think you've got yourself confused between arrays and numbers, specifically what it means to manipulate binary numbers.
I'll go about this by example. Say you have a number of error messages and you want to return them in a return value from a function. Now, you might label your errors 1,2,3,4... which makes sense to your mind, but then how do you, given just one number, work out which errors have occured?
Now, try labelling the errors 1,2,4,8,16... increasing powers of two, basically. Why does this work? Well, when you work base 2 you are manipulating a number like 00000000 where each digit corresponds to a power of 2 multiplied by its position from the right. So let's say errors 1, 4 and 8 occur. Well, then that could be represented as 00001101. In reverse, the first digit = 1*2^0, the third digit 1*2^2 and the fourth digit 1*2^3. Adding them all up gives you 13.
Now, we are able to test if such an error has occured by applying a bitmask. By example, if you wanted to work out if error 8 has occured, use the bit representation of 8 = 00001000. Now, in order to extract whether or not that error has occured, use a binary and like so:
00001101
& 00001000
= 00001000
I'm sure you know how an and works or can deduce it from the above - working digit-wise, if any two digits are both 1, the result is 1, else it is 0.
Now, in C:
int func(...)
{
int retval = 0;
if ( sometestthatmeans an error )
{
retval += 1;
}
if ( sometestthatmeans an error )
{
retval += 2;
}
return retval
}
int anotherfunc(...)
{
uint8_t x = func(...)
/* binary and with 8 and shift 3 plaes to the right
* so that the resultant expression is either 1 or 0 */
if ( ( ( x & 0x08 ) >> 3 ) == 1 )
{
/* that error occurred */
}
}
Now, to practicalities. When memory was sparse and protocols didn't have the luxury of verbose xml etc, it was common to delimit a field as being so many bits wide. In that field, you assign various bits (flags, powers of 2) to a certain meaning and apply binary operations to deduce if they are set, then operate on these.
I should also add that binary operations are close in idea to the underlying electronics of a computer. Imagine if the bit fields corresponded to the output of various circuits (carrying current or not). By using enough combinations of said circuits, you make... a computer.
regarding the usage the bits array :
if you know there are "only" 1 million numbers - you use an array of 1 million bits. in the beginning all bits will be zero and every time you read a number - use this number as index and change the bit in this index to be one (if it's not one already).
after reading all numbers - the missing numbers are the indices of the zeros in the array.
for example, if we had only numbers between 0 - 4 the array would look like this in the beginning: 0 0 0 0 0.
if we read the numbers : 3, 2, 2
the array would look like this: read 3 --> 0 0 0 1 0. read 3 (again) --> 0 0 0 1 0. read 2 --> 0 0 1 1 0. check the indices of the zeroes: 0,1,4 - those are the missing numbers
BTW, of course you can use integers instead of bits but it may take (depends on the system) 32 times memory
Sivan
Bit Arrays or Bit Vectors can be though as an array of boolean values. Normally a boolean variable needs at least one byte storage, but in a bit array/vector only one bit is needed.
This gets handy if you have lots of such data so you save memory at large.
Another usage is if you have numbers which do not exactly fit in standard variables which are 8,16,32 or 64 bit in size. You could this way store into a bit vector of 16 bit a number which consists of 4 bit, one that is 2 bit and one that is 10 bits in size. Normally you would have to use 3 variables with sizes of 8,8 and 16 bit, so you only have 50% of storage wasted.
But all these uses are very rarely used in business aplications, the come to use often when interfacing drivers through pinvoke/interop functions and doing low level programming.
Bit Arrays of Bit Vectors are used as a mapping from position to some bit value. Yes it's basically the same thing as an array of Bool, but typical Bool implementation is one to four bytes long and it uses too much space.
We can store the same amount of data much more efficiently by using arrays of words and binary masking operations and shifts to store and retrieve them (less overall memory used, less accesses to memory, less cache miss, less memory page swap). The code to access individual bits is still quite straightforward.
There is also some bit field support builtin in C language (you write things like int i:1; to say "only consume one bit") , but it is not available for arrays and you have less control of the overall result (details of implementation depends on compiler and alignment issues).
Below is a possible way to answer to your "search missing numbers" question. I fixed int size to 32 bits to keep things simple, but it could be written using sizeof(int) to make it portable. And (depending on the compiler and target processor) the code could only be made faster using >> 5 instead of / 32 and & 31 instead of % 32, but that is just to give the idea.
#include <stdio.h>
#include <errno.h>
#include <stdint.h>
int main(){
/* put all numbers from 1 to 1000000 in a file, except 765 and 777777 */
{
printf("writing test file\n");
int x = 0;
FILE * f = fopen("testfile.txt", "w");
for (x=0; x < 1000000; ++x){
if (x == 765 || x == 777760 || x == 777791){
continue;
}
fprintf(f, "%d\n", x);
}
fprintf(f, "%d\n", 57768); /* this one is a duplicate */
fclose(f);
}
uint32_t bitarray[1000000 / 32];
/* read file containing integers in the range [1,1000000] */
/* any non number is considered as separator */
/* the goal is to find missing numbers */
printf("Reading test file\n");
{
unsigned int x = 0;
FILE * f = fopen("testfile.txt", "r");
while (1 == fscanf(f, " %u",&x)){
bitarray[x / 32] |= 1 << (x % 32);
}
fclose(f);
}
/* find missing number in bitarray */
{
int x = 0;
for (x=0; x < (1000000 / 32) ; ++x){
int n = bitarray[x];
if (n != (uint32_t)-1){
printf("Missing number(s) between %d and %d [%x]\n",
x * 32, (x+1) * 32, bitarray[x]);
int b;
for (b = 0 ; b < 32 ; ++b){
if (0 == (n & (1 << b))){
printf("missing number is %d\n", x*32+b);
}
}
}
}
}
}
That is used for bit flags storage, as well as for parsing different binary protocols fields, where 1 byte is divided into a number of bit-fields. This is widely used, in protocols like TCP/IP, up to ASN.1 encodings, OpenPGP packets, and so on.
I'm a beginner (self-learning) programmer learning C++, and recently I decided to implement a binary-coded decimal (BCD) class as an exercise, and so I could handle very large numbers on Project Euler. I'd like to do it as basically as possible, starting properly from scratch.
I started off using an array of ints, where every digit of the input number was saved as a separate int. I know that each BCD digit can be encoded with only 4 bits, so I thought using a whole int for this was a bit overkill. I'm now using an array of bitset<4>'s.
Is using a library class like this overkill as well?
Would you consider it cheating?
Is there a better way to do this?
EDIT: The primary reason for this is as an exercise - I wouldn't want to use a library like GMP because the whole point is making the class myself. Is there a way of making sure that I only use 4 bits for each decimal digit?
Just one note, using an array of bitset<4>'s is going to require the same amount of space as an array of long's. bitset is usually implemented by having an array of word sized integers be the backing store for the bits, so that bitwise operations can use bitwise word operations, not byte ones, so more gets done at a time.
Also, I question your motivation. BCD is usually used as a packed representation of a string of digits when sending them between systems. There isn't really anything to do with arithmetic usually. What you really want is an arbitrary sized integer arithmetic library like GMP.
Is using a library class like this overkill as well?
I would benchmark it against an array of ints to see which one performs better. If an array of bitset<4> is faster, then no it's not overkill. Every little bit helps on some of the PE problems
Would you consider it cheating?
No, not at all.
Is there a better way to do this?
Like Greg Rogers suggested, an arbitrary precision library is probably a better choice, unless you just want to learn from rolling your own. There's something to learn from both methods (using a library vs. writing a library). I'm lazy, so I usually use Python.
Like Greg Rogers said, using a bitset probably won't save any space over ints, and doesn't really provide any other benefits. I would probably use a vector instead. It's twice as big as it needs to be, but you get simpler and faster indexing for each digit.
If you want to use packed BCD, you could write a custom indexing function and store two digits in each byte.
Is using a library class like this overkill as well?
Would you consider it cheating?
Is there a better way to do this?
1&2: not really
3: each byte's got 8-bits, you could store 2 BCD in each unsigned char.
In general, bit operations are applied in the context of an integer, so from the performance aspect there is no real reason to go to bits.
If you want to go to bit approach to gain experience, then this may be of help
#include <stdio.h>
int main
(
void
)
{
typedef struct
{
unsigned int value:4;
} Nibble;
Nibble nibble;
for (nibble.value = 0; nibble.value < 20; nibble.value++)
{
printf("nibble.value is %d\n", nibble.value);
}
return 0;
}
The gist of the matter is that inside that struct, you are creating a short integer, one that is 4 bits wide. Under the hood, it is still really an integer, but for your intended use, it looks and acts like a 4 bit integer.
This is shown clearly by the for loop, which is actually an infinite loop. When the nibble value hits, 16, the value is really zero, as there are only 4 bits to work with.
As a result nibble.value < 20 never becomes true.
If you look in the K&R White book, one of the notes there is the fact that bit operations like this are not portable, so if you want to port your program to another platform, it may or may not work.
Have fun.
You are trying to get base-10 representation (i.e. decimal digit in each cell of the array). This way either space (one int per digit), or time (4-bits per dgit, but there is overhead of packing/unpacking) is wasted.
Why not try with base-256, for example, and use an array of bytes? Or even base-2^32 with array of ints? The operations are implemented the same way as in base-10. The only thing that will be different is converting the number to a human-readable string.
It may work like this:
Assuming base-256, each "digit" has 256 possible values, so the numbers 0-255 are all single digit values. Than 256 is written as 1:0 (I'll use colon to separate the "digits", we cannot use letters like in base-16), analoge in base-10 is how after 9, there is 10.
Likewise 1030 (base-10) = 4 * 256 + 6 = 4:6 (base-256).
Also 1020 (base-10) = 3 * 256 + 252 = 3:252 (base-256) is two-digit number in base-256.
Now let's assume we put the digits in array of bytes with the least significant digit first:
unsigned short digits1[] = { 212, 121 }; // 121 * 256 + 212 = 31188
int len1 = 2;
unsigned short digits2[] = { 202, 20 }; // 20 * 256 + 202 = 5322
int len2 = 2;
Then adding will go like this (warning: notepad code ahead, may be broken):
unsigned short resultdigits[enough length] = { 0 };
int len = len1 > len2 ? len1 : len2; // max of the lengths
int carry = 0;
int i;
for (i = 0; i < len; i++) {
int leftdigit = i < len1 ? digits1[i] : 0;
int rightdigit = i < len2 ? digits2[i] : 0;
int sum = leftdigit + rightdigit + carry;
if (sum > 255) {
carry = 1;
sum -= 256;
} else {
carry = 0;
}
resultdigits[i] = sum;
}
if (carry > 0) {
resultdigits[i] = carry;
}
On the first iteration it should go like this:
sum = 212 + 202 + 0 = 414
414 > 256, so carry = 1 and sum = 414 - 256 = 158
resultdigits[0] = 158
On the second iteration:
sum = 121 + 20 + 1 = 142
142 < 256, so carry = 0
resultdigits[1] = 142
So at the end resultdigits[] = { 158, 142 }, that is 142:158 (base-256) = 142 * 256 + 158 = 36510 (base-10), which is exactly 31188 + 5322
Note that converting this number to/from a human-readable form is by no means a trivial task - it requires multiplication and division by 10 or 256 and I cannot present code as a sample without proper research. The advantage is that the operations 'add', 'subtract' and 'multiply' can be made really efficient and the heavy conversion to/from base-10 is done only once in the beginning and once after the end of the calculation.
Having said all that, personally, I'd use base 10 in array of bytes and not care about the memory loss. This will require adjusting the constants 255 and 256 above to 9 and 10 respectively.