Express and Plot Complex Numbers
Learning Objectives
- Express square roots of negative numbers as multiples of i
- Plot complex numbers on the complex plane
[latex]\sqrt{-1}=i[/latex]
So, using properties of radicals,[latex]{i}^{2}={\left(\sqrt{-1}\right)}^{2}=-1[/latex]
We can write the square root of any negative number as a multiple of i. Consider the square root of –25.[latex]\begin{array}{l} \sqrt{-25}=\sqrt{25\cdot \left(-1\right)}\hfill \\ \text{ }=\sqrt{25}\sqrt{-1}\hfill \\ \text{ }=5i\hfill \end{array}[/latex]
We use 5i and not [latex]-\text{5}i[/latex] because the principal root of 25 is the positive root. A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a + bi where a is the real part and bi is the imaginary part. For example, [latex]5+2i[/latex] is a complex number. So, too, is [latex]3+4\sqrt{3}i[/latex]. Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.A General Note: Imaginary and Complex Numbers
A complex number is a number of the form [latex]a+bi[/latex] where- a is the real part of the complex number.
- bi is the imaginary part of the complex number.
How To: Given an imaginary number, express it in standard form.
- Write [latex]\sqrt{-a}[/latex] as [latex]\sqrt{a}\sqrt{-1}[/latex].
- Express [latex]\sqrt{-1}[/latex] as i.
- Write [latex]\sqrt{a}\cdot i[/latex] in simplest form.
Example: Expressing an Imaginary Number in Standard Form
Express [latex]\sqrt{-9}[/latex] in standard form.Answer: [latex-display]\sqrt{-9}=\sqrt{9}\sqrt{-1}=3i[/latex-display] In standard form, this is [latex]0+3i[/latex].
Try It
Express [latex]\sqrt{-24}[/latex] in standard form.Answer: [latex]\sqrt{-24}=0+2i\sqrt{6}\\[/latex]
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- Introduction to Complex Numbers. Authored by: Sousa, James. License: CC BY: Attribution.
- Question ID 61706. Authored by: Day, Alyson. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
- Question ID 65709. Authored by: Kaslik,Pete, mb Lippman,David. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].