A First Look At Algebra and Number Sets
This is the start of a new series involving Algebra and number sets. It will start out with basic Algebra but get more advanced in time.
I have been wanting to start this series for a long time. I think advanced Algebra is very interesting and I want to explore it as much as I can. However, I need to start out with the basics for those that would not be familiar with it.
I have been thinking how far in the beginning I wanted to start. It eventually occurred to me that I wanted to start at the very beginning for the sake of my son. My son is young and has no interest in math yet. I hope that changes soon.
This series is supposed to be something that he can eventually follow and learn from if he develops that interest. I will just expose math to him regularly and hope for the best.
The basics of numbers is about patterns. Whenever we look at a series of numbers a pattern often develops. As we go about our day to day activities we use numbers and the resulting patterns to help us do a better job.
Most of the time we do this unconsciously. However, if we think about it a little more we can use this number behavior to our advantage.
First, let us look at numbers from a very basic standpoint so we are all on the same page. Numbers can be rational, irrational, natural or integers.
5 = natural, integer, and rational
-1.2 = rational
13/7 = rational
sqrt -7 = irrational
-12 = integer and rational
sqrt 16 = natural, integer, and rational because it evaluates to 4.
These definitions will help us later on when dealing with certain mathematical situations. Each type of number I have listed above has certain rules about what can be done with them. It is these rules that dictate how we interact with problems to solve them. From here on I will give some basic examples of these points to cement them in your heads.
Example - Find the average of some amount of numbers.
If you have 72 computers with varying numbers on 3 different pallets, how many computers are on each pallet on average?
This is a nice and easy example to start with. Most of you can instantly recognize that the answer is 24. A few might have to bust out a calculator but you understand how to solve the problem at the very least.
Now lets do another example that many adults don’t know how to do but is simple nonetheless. This one is about calculating percent change in two numbers. I asked a couple people and neither adult knew how to do this off top of their heads.
They are both proficient with Excel though. Kind of concerning isn’t it? I think we often rely on our tools too much. In fact, many people can’t do things by hand anymore.
That is also the reason why I get asked to help with these things, as these people don’t know why the calculations they do work or don’t work.
When something does not give the expected results, they can’t debug their own problem to figure out why.
Solve: Calculate the percent change in these two numbers: 38 and 57
I will explain what we are doing so it is clear. First, we get the difference between the two numbers. 57-38 = 19. We then divide the difference by the lesser number. 19/38 = .5. Multiply the .5 times 100 and that is your percent change. .5 * 100 = 50% change.
If you notice, I am going over some common tasks that everyone should know how to do. I am trying to explain how they work also. Another problem someone might be curious about is how fast the Earth moves. So lets work on this.
Solve: Calculate the speed of Earth in its orbit
Let’s think about what we know. Its orbit is the elliptical path it takes around the Sun. An elliptical path is a very stretched out circle in simple terms. The Earth goes around the Sun because of the Sun’s gravitational pull. We also know that the average radius of our orbital path is 93 million miles. The Earth also takes approximately 1 year to travel this path. We will use this simple geometry to figure out the rest of what we need.
Speed is distance divided by time. Our first task is to find out how far we travel in a year. How many miles is it really? We want to use the formula for a circle here because it’s a fair approximation. The distance around a circle is 2* pi * r.
D = 2 * 3.14 * 93,000,000 which is rounded to 584,000,000 miles. Hours in a year = 365 * 24 = 8760.
We have our problem data now so just divide. 584,000,000 / 8760 = 67,000 miles per hour. I rounded that up. That is how fast we are traveling around the Sun.
Another good ability to have is to be able to calculate the volume of common objects. This is handy in cooking, 3d printing, and estimating the mass of stellar objects 50 billion light years away. Fun right! So what would the volume be of a simple soda can?
Solve: Find the volume of the infamous soda can
This soft drink is a cylinder. That is its shape. The formula for the volume of a cylinder is v = pi * r^2 * h. V = volume, pi = 3.14, r = radius, and h = height. If r = 1.4 inches and and h = 5 inches, use these figures to find the volume.
V = (3.14) * (1.4)^2 * (5).
V = 30.78 cubic inches.
Another good tool to have in your mental repertoire is being able to measure the thickness of objects through basic calculations. It’s actually a pretty quick calculation once you understand it. Lets try it out.
Solve: What is the thickness of an object that is 15 cm by 35 cm and weighs 5.4 grams
Thickness = volume / area. 1 cubic cm of this material weighs 2.7 grams. We start by finding the volume. Volume in this case will be length * width * thickness.
We know the material weighs a total of 5.4 grams and also that 1 cubic cm is equal to 2.7 grams. So that means we have 2 cubic cm of material, which is our volume.
The area of this material is easy since it seems to be rectangular. 15 cm * 35 cm = 525 cm ^2.
Thickness = volume / area. Thickness = 2 cm ^3 / 525 cm ^2. Thickness = .0038 cm. That is our answer. Its pretty thin isn’t it? This process does show you how useful it is to be able to do these calculations. You have to have the correct information up front of course.
I am going to stop here for this entry. In this topic I talked about the types of numbers we often see. I hopefully explained why it is important to do calculations by hand at first, until they are understood.
We did several examples together and I hope they were able to be understood by my explanations. This will be an ongoing series and I think it will be a lot of fun to learn and teach others. Percy, if you read this far, this one was for you my son!