A "complex number" is a number of the form a+bi, where a and b are both real numbers and i is the principal square root of -1.
Since b can be equal to 0, you see that the real numbers are a subset of the complex numbers. Similarly, since a can be zero, the imaginary numbers are a subset of the complex numbers.
So let's take two complex numbers: a+bi and c+di (where a, b, c, and d are real). We add them together and we get:
(a+c) + (b+d)i
The sum of two real numbers is always real, so a+c is a real number and b+d is a real number, so the sum of two complex numbers is a complex number.
What you may really be wondering is whether the sum of two non-real complex numbers can ever be a real number. The answer is yes:
(3+2i) + (5-2i) = 8.
In fact, the complex numbers form an algebraic field. The sum, difference, product, and quotient of any two complex numbers (except division by 0) is a complex number (keeping in mind the special case that both real and imaginary numbers are a subset of the complex numbers).
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The sum of any two prime numbers is not always a composite number. The sum of 2 and 11 is 13, and 13 is a prime number, not a composite number.
Lots of numbers do. To begin, all real numbers do. Multiples of sqrt(-1), aka. imaginary numbers, do. The Complex Numbers are all numbers which are the sum of a real number and an imaginary number.
Yes.
Yes.
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