Every real number is an integer?

Integers are a subset of real numbers. All integers are real numbers, but not the other way around.

Real numbers, without going into great detail, are numbers that can be drawn on a standard x-y graph. This includes integers, fractions & decimals, common constants such as pi and e, etc. Taking the square root of negative number would result in an imaginary number (which are not a real numbers.)

Integers are essentially whole numbers (numbers that can be written without a decimal or in fraction form).