Arithmetic with Complex Numbers: IM Alg2.3.12
[url=https://im.kendallhunt.com/HS/teachers/3/3/12/preparation.html]“Arithmetic with Complex Numbers”[/url] from IM Algebra II © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg2.3.12 Lesson
- IM Alg2.3.12 Lesson: Arithmetic with Complex Numbers
- Alg2.3.12 Practice
- IM Alg2.3.12 Practice: Arithmetic with Complex Numbers
IM Alg2.3.12 Lesson: Arithmetic with Complex Numbers
Find the value of these expressions mentally.
[math]2-2+20-20+200-200[/math]
[math]100-50+10-10+50-10[/math]
[math]3+2+1+0-1-2-3[/math]
[math]1+2+4+8+16+32-16-8-4-2-1[/math]
This diagram represents (2+3i)+(-8-8i).
How do you see [math]2+3i[/math] represented?
How do you see [math]\text{-}8-8i[/math] represented?[br]
What complex number does [math]A[/math] represent?[br]
Add “like terms” in the expression [math](2+3i)+(\text{-}8-8i)[/math]. What do you get?[br]
[size=100][size=150]Write these sums and differences in the form [math]a+bi[/math], where [math]a[/math] and [math]b[/math] are real numbers.[/size][/size][br][math](\text{-}3+2i)+(4-5i)[/math] (Check your work by drawing a diagram in the applet below.)[br]
[math](\text{-}37-45i)+(11+81i)[/math]
[math](\text{-}3+2i)-(4-5i)[/math]
[math](\text{-}37-45i)-(11+81i)[/math]
Drag the points to represent 2, 2², 2³, and 2⁴ on the real number line.
Write [math]2i[/math] in the form [math]a+bi[/math].
Write [math](2i)^2[/math] in the form [math]a+bi[/math].
Write [math](2i)^3[/math] in the form [math]a+bi[/math].[br]
Write [math](2i)^4[/math] in the form [math]a+bi[/math].[br]
[size=150]Plot [math]2i[/math], [math](2i)^2[/math], [math](2i)^3[/math], and [math](2i)^4[/math] on the complex plane.[/size]
If [math]a[/math] and [math]b[/math] are positive numbers, is it true that [math]\sqrt{ab}=\sqrt{a}\sqrt{b}[/math]? Explain how you know.[br]
If [math]a[/math] and [math]b[/math] are negative numbers, is it true that [math]\sqrt{ab}=\sqrt{a}\sqrt{b}[/math]? Explain how you know.
IM Alg2.3.12 Practice: Arithmetic with Complex Numbers
Write each expression in the form a+bi, where a and b are real numbers.
You may plot the numbers in the complex plane as a guide.[br][list][*][math]2\cdot\sqrt{\text{-}4}[/math][br][/*][*][math]3i\cdot2i[/math][/*][*][math]i^4[/math][/*][*][math]4-3\sqrt{\text{-}1}[/math][/*][/list]
[size=150]Which expression is equivalent to [math](3+9i)-(5-3i)[/math]?[/size]
[size=150]What are [math]a[/math] and [math]b[/math] when you write[math]\sqrt{\text{-}16}[/math] in the form [math]a+bi[/math], where [math]a[/math] and [math]b[/math] are real numbers?[/size]
Fill in the boxes to make a true statement:
Plot each number on the real number line, or explain why the number is not on the real number line.
[list][*][math]\sqrt{16}[/math][/*][*][math]-\sqrt{16}[/math][/*][*][math]\sqrt{\text{-}16}[/math][/*][*][math]56^{\frac{1}{2}}[/math][/*][*][math]-56^{\frac{1}{2}}[/math][/*][*][math](-56)^{\frac{1}{2}}[/math][/*][/list]
[size=150]Which expression is equivalent to [math]\sqrt{\text{-}4}[/math]?[/size]