Why should I do cin.clear() before cin.ignore() to flush cin?

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C++

Attached is my programming homework. I was trying to deal with inputs that are neither expected integers nor EOF. I noticed that to avoid infinite loops I had to flush out cin before reading next input, but:

1. I'm not entirely sure why. Is it because a wrong input is just put back into the input stream?

2. Why is cin.ignore() alone not enough to stop infinite loops?

3. What confuses me most is I can't tell the difference between
cin.clear() then cin.ignore() and
cin.ignore() then cin.clear()
Apparently only the former order works. But what does stream error state have to do with dumping a stream?

#include<iostream>
#include<iomanip>

using namespace std;

int main()
{
	int num;
	
	cout << "Please enter a decimal number: ";
	cin >> num;

	while (!cin.eof())
	{
		if (!cin.fail())
		{
			cout << "Its hexadecimal format is: " << setbase(16) << num << endl;
			cout << "Its octal format is: " << setbase(8) << num << endl;
		}
		cout << "Please enter a decimal number: ";
		cin.clear();
		cin.ignore(numeric_limits<streamsize>::max(), '\n');
		cin >> num;
	}

	return 0;
}