Some compilers, such as GCC, offer a "long long" type, giving 64 bits capable of representing about 9 quintillion (9 times 1018)
This is good for most purposes, but some applications require many more digits than this. For example, public-key encryption with the RSA algorithm typically requires 300 digit numbers. Computing the probabilities of certain real events often involves very large numbers; although the result might fit in a typical C type, the intermediate computations require very large numbers.
For example, what is the probability of winning the Texas Lottery jackpot prize with one ticket? The number of combinations of 50 numbers taken 6 at a time, "50 choose 6", is 50!/((50-6)!6!). That number is 15,890,700, so the odds of winning are 1/15,890,700. The number 15,890,700 can be represented easily by a C integer, but the (naive) computation of that number involves computing 50!, which is:
30,414,093,201,713,378,043,612,608,166,064,768,844,377,641,568,960,512,000,000,000,000This number will not fit into a C integer, not even a 64 bit one.
So we must move to a different representation of non-negative integers. We can represent a number as a sequence of digits stored in an array of integers. We can write functions to add, multiply, etc. on those arrays, and then make them as large as we want.
In our new representation, we have an array of "digits" (integers) in some base b. Typically, b = 10, our normal decimal number system; that makes things easy to print. The 0'th array element is the 1's place, the #1 element is the ten's place, the #2 element is the hundred's place, and so forth (really, the b0's place, the b1 place, the b2 place, etc.)
Let's look at some algorithms doing arithmetic on our new "big" integers. We'll let N be the number of digits we will represent. If we need any more, we just increase N. Let BASE be the base of our number system, understood to be 10 typically, but can be changed if we like.
First, we need a way of making a "normal" integer into a "big" integer; we'd like a function called make_int such that e.g. make_int (A, 123) would put 3 in A[0], 2 in A[1], 1 in A[2], and zeros in A[3..N-1]. We'll do these in C rather than pseudocode because the code works out very easily:
/* put the normal int n into the big int A */
void make_int (int A[], int n) {
int i;
/* start indexing at the 0's place */
i = 0;
/* while there is still something left to the number
* we're encoding... */
while (n) {
/* put the least significant digit of n into A[i] */
A[i++] = n % BASE;
/* get rid of the least significant digit,
* i.e., shift right once
*/
n /= BASE;
}
/* fill the rest of the array up with zeros */
while (i < N) A[i++] = 0;
}
This algorithm takes 
(N)
time and space.
Now let's look at an algorithm to add one to a big integer. This is a common operation and easier than full addition, so we'll look at it first:
/* A++ */
void increment (int A[]) {
int i;
/* start indexing at the least significant digit */
i = 0;
while (i < N) {
/* increment the digit */
A[i]++;
/* if it overflows (i.e., it was 9, now it's 10, too
* big to be a digit) then...
*/
if (A[i] == BASE) {
/* make it zero and index to the next
* significant digit
*/
A[i] = 0;
i++;
} else
/* otherwise, we are done */
break;
}
}
This algorithm takes O(N) time in the worst case
(imagine 9999999999...) and 
(1)
in the best case (no overflow in the least significant digit).
Now let's look at the more general case of addition of two big integers. Here, we want to add two big ints in arrays called A[0..N-1] and B[0..N-1], and put the result into C[0..N-1]. We'll use the algorithm we learned in grade school: add corresponding digits, plus a "carry" generated by previous overflows.
/* C = A + B */
void add (int A[], int B[], int C[]) {
int i, carry, sum;
/* no carry yet */
carry = 0;
/* go from least to most significant digit */
for (i=0; i<N; i++) {
/* the i'th digit of C is the sum of the
* i'th digits of A and B, plus any carry
*/
sum = A[i] + B[i] + carry;
/* if the sum exceeds the base, then we have a carry. */
if (sum >= BASE) {
carry = 1;
/* make sum fit in a digit (same as sum %= BASE) */
sum -= BASE;
} else
/* otherwise no carry */
carry = 0;
/* put the result in the sum */
C[i] = sum;
}
/* if we get to the end and still have a carry, we don't have
* anywhere to put it, so panic!
*/
if (carry) printf ("overflow in addition!\n");
}
This function does constant work in a loop that iterates N
times, so the time for addition is (N).
Multiplication is next. Recall from grade school how you multiplied two large numbers A and B: starting with the least significant digit, you multiplied each digit of A with every digit of B, forming a partial product. You shifted this product over to the left for each new digit, writing overflowed digits above A to remind yourself to add them in. We will need a function multiply_one_digit that will multiply an entire big integer by a single digit, placing the result in a new big int. We also need a function shift_left that shift a number over to the left a number of spaces, effectively multiplying it by BASEi where i is the number of spaces. Here is the algorithm to multiply:
/* C = A * B */
void multiply (int A[], int B[], int C[]) {
int i, j, P[N];
/* C will accumulate the sum of partial products. It's initially 0. */
make_int (C, 0);
/* for each digit in A... */
for (i=0; i<N; i++) {
/* multiply B by digit A[i] */
multiply_one_digit (B, P, A[i]);
/* shift the partial product left i bytes */
shift_left (P, i);
/* add result to the running sum */
add (C, P, C);
}
}
Now let's look at the function that multiplies by
a single digit:
/* B = n * A */
void multiply_one_digit (int A[], int B[], int n) {
int i, carry;
/* no extra overflow to add yet */
carry = 0;
/* for each digit, starting with least significant... */
for (i=0; i<N; i++) {
/* multiply the digit by n, putting the result in B */
B[i] = n * A[i];
/* add in any overflow from the last digit */
B[i] += carry;
/* if this product is too big to fit in a digit... */
if (B[i] >= BASE) {
/* handle the overflow */
carry = B[i] / BASE;
B[i] %= BASE;
} else
/* no overflow */
carry = 0;
}
if (carry) printf ("overflow in multiplication!\n");
}
And finally the function to shift left a certain number of spaces:
/* "multiplies" a number by BASEn */
void shift_left (int A[], int n) {
int i;
/* going from left to right, move everything over to the
* left n spaces
*/
for (i=N-1; i>=n; i--) A[i] = A[i-n];
/* fill the last n digits with zeros */
while (i >= 0) A[i--] = 0;
}
The shift_left and multiply_one_digit algorithms
each do a constant amount of work in a loop that runs for
(N) time, so they each
take time (N).
Addition also takes (N)
time; all three are done in the multiply function a constant
number of times within a loop that iterates N times, so
multiply takes a time in
(N2).
Some comments on large number arithmetic:
(Nlg 3)
= O(N1.59), much better than the
quadratic time above. However, this technique only becomes efficient for
very large values of N. There is another technique using the
Fast Fourier Transform that multiples numbers in
O(N log N log log N)
time, which is even better, but still only becomes efficient for large
values of N (e.g. > 10,000 decimal digits).