In the quadratic formula,
If , then it has two real roots that are different from one another. If , then it has just one real root. This root is sometimes called a double root, since it can be considered a root that is repeated. If , then it has two complex roots that are different from one another.
An imaginary number is any number that can be written in the form, where is a real number and . We can find imaginary numbers by taking the square roots of negative numbers. Since the imaginary number is defined by the equation , we can write all square roots of negative numbers in terms of . For example, , and . The set of imaginary numbers lies completely separate from the set of real numbers.
A complex number is any number that can be written in the form, where and are real numbers and . If , this will give us an imaginary number, and if , this will give us a real number. The set of complex numbers full encompasses the real numbers and the complex numbers, but it also has members that are neither real nor imaginary. In other words, all real numbers are complex, and all imaginary numbers are complex. Some complex numbers, however, have both imaginary and real parts, preventing them from falling into the imaginary category or the real category. has a real part and an imaginary part , so it is a complex number. Although all imaginary numbers and all real numbers are complex numbers, when a number is specifically referred to as a complex number, this sometimes means that it has both real and imaginary parts (i.e. it is neither real nor imaginary).
We can use the complex plane to visualize complex numbers, just as we can use the number line to visualize real numbers. The complex plane is formed by two perpendicular axes. The horizontal one represents the real numbers, just like a real number line, and the vertical one represents the imaginary numbers. Any complex number of the formcan be plotted on the complex plane by letting its horizontal coordinate be equal to its real part, , and its vertical coordinate be equal to the coefficient of its imaginary part , . We can find the number at the point , and we can find the number at the point .
Complex conjugates are complex numbers whose real parts are equal and whose imaginary parts have opposite signs. So if one complex number is
*The complex conjugate of is .
*The complex conjugate of is . (Remember, all imaginary numbers are complex numbers!)
*The complex conjugate of is . (Remember, all real numbers are complex numbers!)