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Imaginary Unit.
Imaginary Unit.
We define the imaginary unit, [latex]i[/latex], by
[latex]{i^2=-1}~~~~~~\text{or}~~~~~~{i=\sqrt{-1}}[/latex]
Chapter 10: Polar Coordinates and Complex Numbers
Solve.
[latex]\underline{\qquad\qquad\qquad\qquad}[/latex]
Learning Objectives
So far most of your work in mathematics has been done using the set of real numbers. We often represent the real numbers by a number line because they can be matched up one-for-one with the points on the line. Every real number is either rational or irrational and can be expressed as a decimal number, although irrational numbers are non-repeating, non-terminating decimals. However, the real numbers are actually a subset of a larger set of numbers called the complex numbers.
You may have first encountered complex numbers as solutions of certain quadratic equations. For example, the graph of
[latex]f(x)=x^2-2x+2[/latex]
has no [latex]x[/latex]-intercepts (as shown at right) because the equation [latex]~~x^2-2x+2-0~~[/latex] has no real-valued solutions.
Applying the quadratic formula, we find
[latex]x=\dfrac{-(-2) \pm \sqrt{(-2)^2-4(1)(2)}}{2} = \dfrac{2 \pm \sqrt{-4}}{2}[/latex]
The solutions of the equation are [latex]\dfrac{2 + \sqrt{-4}}{2}[/latex] and [latex]\dfrac{2 - \sqrt{-4}}{2}\text{,}[/latex] but they are not real numbers. Because [latex]\sqrt{-4}[/latex] is not a real number, the equation [latex]x^2-2x+2-0[/latex] has no real solutions.
Although square roots of negative numbers such as are not real numbers, they occur often in mathematics and its applications. Mathematicians began working with square roots of negative numbers in the sixteenth century in their attempts to solve quadratic and cubic equations. René Descartes gave them the name imaginary numbers, which reflected the mistrust with which mathematicians regarded them at the time. Today, however, such numbers are well understood and used routinely by scientists and engineers.
We begin by defining a new number, called [latex]i[/latex], whose square is [latex]-1\text{.}[/latex]
We define the imaginary unit, [latex]i[/latex], by
[latex]{i^2=-1}~~~~~~\text{or}~~~~~~{i=\sqrt{-1}}[/latex]
The letter [latex]i[/latex] used in this way is not a variable; it is the name of a specific number, and hence is a constant. The square root of any negative number can be written as the product of a real number and [latex]i[/latex]. For example,
[latex]\sqrt{-4} = \sqrt{-1 \cdot 4}\\ = \sqrt{-1} \sqrt{4} = i \cdot 2[/latex]
or [latex]\sqrt{-4} = 2\text{.}[/latex] Any number that is the product of [latex]i[/latex][latex]i[/latex] and a real number is called an imaginary number.
For any real number [latex]a \gt 0\text{,}[/latex]
[latex]{\sqrt{-a} = \sqrt{-1}\sqrt{a} = i\sqrt{a}}[/latex]
Examples of imaginary numbers are [latex]3i,~ \dfrac{7}{8}i,~ -38i,[/latex] and [latex]i\sqrt{5}\text{.}[/latex]
Write each radical as an imaginary number.
Write each radical as an imaginary number.
Just as each positive number has two real-valued square roots, every n