The Complex Numbers The ratio of the length of a diagonal of a square to the length of a side cannot be represented as the quotient of two integers.

Numbers that are used for measurement and that can be represented on a continuous number line belong to the real number system. The real numbers consist of zero and all positive and negative integers, rational numbers, and irrational numbers. One of the basic properties of real numbers is that their squares are never negative.

Complex Numbers We define the imaginary unit i with the following properties: We then define the square root of any negative number as follows: Even though there is a definition for the square root of a negative number, we cannot assume that all the square root properties of positive numbers will also be true for negative real numbers.

Any number in the form a + bi, where a and b are real numbers and i is the imaginary unit, is called a complex number. In a + bi, a is called the real part and b is called the imaginary part of the complex number. Two complex numbers a + bi and c + di are equal if and only if a = c and b = d.

Complex Conjugates Their sum is a real number, and their product is a nonnegative real number.

Multiply the numerator and the denominator by the conjugate of the denominator.