Why negative values appear sporadically in Fibonacci Series' calculation?

Question

I have coded a program which calculates Fibonacci Series in C++. Here's the code below:

#include <iostream>
using namespace std;

int main() {
    int n, t1 = 0, t2 = 1, nextTerm = 0;

    cout << "Enter the number of terms: ";
    cin >> n;
    cout << endl;

    cout << "Fibonacci Series: ";

    for (int i = 1; i <= n; ++i) {
        // Prints the first two terms.
        if(i == 1) {
            cout << t1 << ", ";
            continue;
        }
        if(i == 2) {
            cout << t2 << ", ";
            continue;
        }
        nextTerm = t1 + t2;
        t1 = t2;
        t2 = nextTerm;

        cout << nextTerm << ", ";
    }
    cout << endl;
    return 0;
}

When I give input n = 100, the list grows. And there appears some negative number.

I know it happens because of the data type value limit. But my question is why some of the are negative and some has positive value.
Like
-285007387, -945834654, -1230842041, 2118290601, 887448560, -1289228135, -401779575, -1691007710,
this is an example from the middle somewhere in the output.

Enter the number of terms: 100
Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, -1323752223, 512559680, -811192543, -298632863, -1109825406, -1408458269, 1776683621, 368225352, 2144908973, -1781832971, 363076002, -1418756969, -1055680967, 1820529360, 764848393, -1709589543, -944741150, 1640636603, 695895453, -1958435240, -1262539787, 1073992269, -188547518, 885444751, 696897233, 1582341984, -2015728079, -433386095, 1845853122, 1412467027, -1036647147, 375819880, -660827267, -285007387, -945834654, -1230842041, 2118290601, 887448560, -1289228135, -401779575, -1691007710, -2092787285, 511172301, -1581614984, -1070442683, 1642909629, 572466946, -2079590721, -1507123775, 708252800, -798870975, -90618175, -889489150, 
Process finished with exit code 0

Answer 1

The appearance of both positive and negative numbers in your Fibonacci series is due to integer overflow.

When an integer overflows, its value wraps around. In the case of signed integers, which typically use two's complement representation, when the value exceeds the maximum positive value representable by the data type, it "wraps around" to the most negative value and continues decreasing. This explains why you're seeing negative values after reaching large positive ones.

For example, with a 32-bit signed integer:

• When you exceed the maximum positive value (2,147,483,647), the value wraps around to the most negative value (-2,147,483,648).
• It then continues decreasing towards more negative values until it reaches the minimum representable value (-2,147,483,648), after which it would overflow again and wrap around to the maximum positive value, creating a cycle of positive to negative values.

That's why in your Fibonacci series, after reaching large positive numbers that caused overflow, you start seeing negative values, and the sequence continues with alternating positive and negative numbers as the overflow wraps around.

Answer 2

As you know, the negative numbers's appearances are caused by data type value limit.
Let's consider the data type that you used in your code, int. The data type int's value limit is from -2^31 to 2^31-1, equals to from -2147483648 to 2147483647.
When a value of type int overflows out of the range, it would be increased or decreased by multiple of the range span of the range, equals to 2^32, in order to keep the value in the range of int.
For example, if you put 2147483648 into a variable of type int, the actual value in that variable is 2147483648-2^32, equals to -2147483648.
Now let's consider the example the output:
-285007387, -945834654, -1230842041, 2118290601, 887448560, -1289228135, -401779575, -1691007710
The third number equals to the sum of the previous two numbers, -285007387and-945834654, equals to (-285007387)+(-945834654)=-1230842041, and that number is in the range, so the third number is -1230842041. The fourth number equals to (-945834654)+(-1230842041)=-2176676695, but that number is out of the range, so we increase it by 2^32, so it becomes (-2176676695)+2^32=2118290601.

After calculating the example output, we can see that the output satisfy this law that ensure the value is in the type int's value range. And because it is only to ensure it is in the range, there is actually no special reasons or methods to know it is positive or negative. In other words, there is no special reasons other than calculation that determine the positivity or negativity of the value.