Why is the "f" required when declaring floats?

Your declaration of a float contains two parts:

1. It declares that the variable timeRemaining is of type float.
2. It assigns the value 0.58 to this variable.

The problem occurs in part 2.

The right-hand side is evaluated on its own. According to the C# specification, a number containing a decimal point that doesn't have a suffix is interpreted as a double.

So we now have a double value that we want to assign to a variable of type float. In order to do this, there must be an implicit conversion from double to float. There is no such conversion, because you may (and in this case do) lose information in the conversion.

The reason is that the value used by the compiler isn't really 0.58, but the floating-point value closest to 0.58, which is 0.57999999999999978655962351581366... for double and exactly 0.579999946057796478271484375 for float.

Strictly speaking, the f is not required. You can avoid having to use the f suffix by casting the value to a float:

float timeRemaining = (float)0.58;

Because there are several numeric types that the compiler can use to represent the value 0.58: float, double and decimal. Unless you are OK with the compiler picking one for you, you have to disambiguate.

The documentation for double states that if you do not specify the type yourself the compiler always picks double as the type of any real numeric literal:

By default, a real numeric literal on the right side of the assignment operator is treated as double. However, if you want an integer number to be treated as double, use the suffix d or D.

Appending the suffix f creates a float; the suffix d creates a double; the suffix m creates a decimal. All of these also work in uppercase.

However, this is still not enough to explain why this does not compile:

float timeRemaining = 0.58;

The missing half of the answer is that the conversion from the double 0.58 to the float timeRemaining potentially loses information, so the compiler refuses to apply it implicitly. If you add an explicit cast the conversion is performed; if you add the f suffix then no conversion will be needed. In both cases the code would then compile.