Glossary for Lesson 16: Complex Numbers


In the quadratic formula, x=\frac {-b\pm \sqrt{b^2-4ac}}{2a} , the expression underneath the square root sign, b^2-4ac , is called the discriminant of the quadratic equation ax^2+bx+c.  The discriminant gives us information about the number and type of roots that a quadratic equation has.
If b^2-4ac>0 , then it has two real roots that are different from one another. If b^2-4ac=0 , 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 b^2-4ac<0 , then it has two complex roots that are different from one another.

Imaginary Numbers

An imaginary number is any number that can be written in the form bi , where b is a real number and i=\sqrt{-1} .  We can find imaginary numbers by taking the square roots of negative numbers.  Since the imaginary number i is defined by the equation i^2=-1 , we can write all square roots of negative numbers in terms of i .  For example, \sqrt{-25}=-5i , and \sqrt {-3} = i\sqrt 3 .  The set of imaginary numbers lies completely separate from the set of real numbers.

Complex Numbers

A complex number is any number that can be written in the form a+bi , where a and b are real numbers and i=\sqrt{-1} .  If a=0 , this will give us an imaginary number, and if b=0 , 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.  3-7i has a real part (3) and an imaginary part (-7i) , 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).

Complex Plane

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 form a+bi can be plotted on the complex plane by letting its horizontal coordinate be equal to its real part, a , and its vertical coordinate be equal to the coefficient of its imaginary part , b.  We can find the number 5-11i at the point (5,-11) , and we can find the number 8+2i at the point (8,2) .

Complex Conjugates

Complex conjugates are complex numbers whose real parts are equal and whose imaginary parts have opposite signs.  So if one complex number is a+bi , then its complex conjugate is a-bi .
*The complex conjugate of -7+6i is -7-6i.
*The complex conjugate of 2i is -2i.  (Remember, all imaginary numbers are complex numbers!)
*The complex conjugate of -9 is -9 .  (Remember, all real numbers are complex numbers!)