Evidently, complex numbers fill the entire two-dimensional plane. The plane is often called the complex plane , and representing complex numbers in this way is sometimes referred to as an Argand Diagram. Visualizing the complex numbers as two-dimensional vectors, it is clear how to add two of them together.
Introduction to Complex Numbers
Multiplying two complex numbers together does not have quite such a simple interpretation. In fact, this representation leads to a clearer picture of multiplication of two complex numbers:.
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In fact this circle — called the unit circle — plays an important part in the theory of complex numbers. Every point on the circle has the form. Bottom Line : any complex number can be written:. What are the square roots of 4?
Simplify roots of negative numbers
That is why we say that the real numbers are a subset of the complex numbers. The complex or imaginary numbers are an extension of the real numbers, characterized by the fact that they give all the roots of the polynomials. This is to say, for any polynomial with real coefficients, it will always have all the solutions in the set of complex numbers.
This is to say, for any polynomial with real coefficients, it will always have all the solutions in the set of complex numbers. Note that in the complex numbers we do not have a total order, as we do with real numbers. This means that when we have two real numbers we can always tell which is the greatest.
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