Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » Advanced Algebra II: Teacher's Guide » Me, Myself, and the Square Root of i

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Comments:

    "This is the "teacher's guide" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Busbee's Math Materials display tagshide tags

    This collection is included inLens: Busbee's Math Materials Lens
    By: Kenneth Leroy Busbee

    Click the "Busbee's Math Materials" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Me, Myself, and the Square Root of i

Module by: Kenny M. Felder. E-mail the author

Summary: A teacher's guide to the square root of the imaginary number.

This is arguably the most advanced, difficult thing we do all year. But I like it because it contains absolutely nothing they haven’t already done. It’s not here because it’s terribly important to know ii size 12{ sqrt {i} } {}, or even because it’s terribly important to know that all numbers can be written in a + b i a+bi format. It is here because it reinforces certain skills—squaring out a binomial (always a good thing to practice), working with variables and numbers together, setting two complex numbers equal by setting the real part on the left equal to the real part on the right and ditto for the imaginary parts, and solving simultaneous equations.

Explain the problem we’re going to solve, hand it out, and let them go. Hopefully, by the end of class, they have all reached the point where they know that 12+ 12i12 size 12{ { {1} over { sqrt {2} } } } {}+12 size 12{ { {1} over { sqrt {2} } } } {}i and 1212i 12 size 12{ { {1} over { sqrt {2} } } } {}12 size 12{ { {1} over { sqrt {2} } } } {}i are the two answers, and have tested them.

Note that right after this in the workbook comes a more advanced version of the same thing, where they find 1313 size 12{ nroot { size 8{3} } { - 1} } {} (all three answers: –1 –1, 1 2 + 32i 1 2 +32 size 12{ { { sqrt {3} } over {2} } } {}i, and 1 2 32i 1 2 32 size 12{ { { sqrt {3} } over {2} } } {}i). I tried using this for the whole class, and it was just a bridge too far. But you could give it to some very advanced students—either as an alternative to the ii size 12{ sqrt {i} } {} exercise, or as an extra credit follow-up to it.

Homework:

They should finish the worksheet if they haven’t done so, including #7. Then they should also do the “Homework: Quadratic Equations and Complex Numbers.” It’s a good opportunity to review quadratic equations, and to bring in something new! (It’s also a pretty short homework.)

When going over the homework, make sure they did #3 by completing the square—again, it’s just a good review, and they can see how the complex answers emerge either way you do it. #4 is back to the discriminant, of course: if b 2 - 4 a c < 0 b 2 -4ac<0 then you will have two complex roots. The answer to #6 is no. The only way to have only one root is if that root is 0. (OK, 0 is technically complex…but that’s obviously not what the question meant, right?)

The fun is seeing if anyone got #5. The answer, of course, is that the two roots are complex conjugates of each other—real part the same, imaginary part different sign. This is obvious if you rewrite the quadratic formula like this:

x = b2a± b24ac2ax=b2a size 12{ { { - b} over {2a} } } {}±b24ac2a size 12{ { { sqrt {b rSup { size 8{2} } - 4 ital "ac"} } over {2a} } } {}

and realize that the part on the left is always real, and the part on the right is where you get your i i from.

Time for another test!

Not much to say here, except that you may want to reuse this extra credit on your own test—if they learn it from the sample (by asking you) and then get it right on your test, they learned something valuable.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks