Algebra 2: Imaginary and Complex Numbers

Happy Wednesday and back to school season math friends! This post introduces imaginary and complex numbers when raised to any power exponent and when multiplied together as a binomial. When it comes to all types of learners, we got you between the video, blog post, and practice problems below. Happy calculating! 🙂

What are Imaginary Numbers?

Imaginary numbers happen when there is a negative under a radical and looks something like this:

Why does this work?

In math, we cannot have a negative under a radical because the number under the square root represents a number times itself, which will always give us a positive number.

Example:

But wait, there’s more:

When raised to a power, imaginary numbers can have the following different values:

Knowing these rules, we can evaluate imaginary numbers, that are raised to any value exponent! Take a look below:

-> We use long division, and divide our exponent value 54, by 4.

-> Now take the value of the remainder, which is 2, and replace our original exponent. Then evaluate the new value of the exponent based on our rules.

What are Complex Numbers?

Complex numbers combine imaginary numbers and real numbers within one expression in a+bi form. For example, (3+2i) is a complex number. Let’s evaluate a binomial multiplying two complex numbers together and see what happens:

-> There are several ways to multiply these complex numbers together. To make it easy, I’m going to show the Box method below:

Try mastering imaginary and complex numbers on your own with the questions below!

Practice:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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COVID-19: What does #FlattenTheCurve even mean?

If you are a human on Earth, then I’m sure you’ve heard about the coronavirus and are currently social distancing. Here in NYC, I’m quarantining like everyone else and listening to all the beautiful math language that has suddenly become mainstream (so, exciting)!  #FlattenTheCurve has become NY’s new catchphrase and for anyone confused about what that means, you’ve come to the right place!

The coronavirus spreads at an Exponential Rate, which means it spreads in a way that increases faster and faster every day.

What does this mean?

For Example, one person with the virus can easily spread the virus to 5 other people, those 5 people can then spread the virus to another 5 people each for a total of an extra 25 people, these 25 people can then spread it to another 5 people each for an extra 125 infected people! And the pattern continues……. See below to get a clearer picture:

Screen Shot 2020-04-12 at 1.26.39 PMScreen Shot 2020-04-12 at 1.20.22 PM.   *Note: These numbers are not based on actual coronavirus data

The Example we just went over is equal to the exponential equation Screen Shot 2020-04-12 at 1.21.48 PM, but it is only that, an Example! The exact pattern and exponential equation of the future progress of the virus is unknown! We mathematicians, can only measure what has already occurred and prepare/model for the future.  To make the virus spread less rapidly, it is our duty to stay home to slow the rate of this exponentially spreading virus as much as possible.

We want to #FlattenTheCurve a.k.a flatten the increasing exponential curve of new coronavirus cases that appear every day! Hopefully, this post brings some clarity to what’s going on in the world right now.  Even with mathematics, the true outcome of the virus may be unknown, but understanding why we are all at home in the first place and the positive impact it has is also important (and kind of cool).

Stay safe math friends and happy mathing! 🙂

Also, please don’t forget to follow more Mathsux on Twitter and Facebook!

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Bored and Confused?

Calling all students, teachers, and parents!  As everyone is stuck at home during a global pandemic, now is a great time we are all forced to try and understand math (and our sanity level) a little bit more.  Well, I may not be able to help you with the keeping sanity stuff, but as far as math goes, hopefully, the below websites offer some much needed mathematic support.

All jokes aside I hope everyone is staying safe and successfully social distancing.  Stay well, math friends! 🙂

Kahn Academy: The same Kahn Academy we know and love still has amazing videos and tutorials as usual, but now they also have a live “homeroom” chat on Facebook LIVE every day at 12:00pm. The chats occur daily with Kahn Academy founder Sal and at times feature famous guests such as Bill Gates. Click the link below for more:

Khan Academy Homeroom Screen Shot 2020-04-04 at 12.21.20 PM

 

Study.com: In a time when companies are being more generous, Study.com is here for us as they offer up to 1000 licenses for school districts and free lessons for teachers, students, and parents.  Check out all the education freebies here:

Study.com

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Math PlanetIf you’re looking for free math resources in Pre-Algebra, Algebra, Algebra 2, and Geometry then you will find the answers you need at Math Planet.  All free all the time, find their website here:

MathPlanet 

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JMAP: For anyone who has to take the NYS Regents at some point (whenever we’re allowed to go outside again), JMAP has every old Regents exam as well as answers to boot! Did I mention each exam is free and printable?  Find their website here:

JMAP

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What is your favorite educational site?  Let me know in the comments, and stay well! 🙂