Arithmetic introduction
- Cliff Stoll
The general approach to what 'math' is, is abstraction. Often times in math, we will take something that very much based in reality, generalize and abstract the concept, sometimes even make up something new that is consistent with the old rules that we were following, and through this abstraction, we can apply the concept to a new area that we couldn't before.
For example (and we will go over the specifics of this in more detail later), many people are familiar with the concept of addition and subtraction. You have \(4\) apples, and I give you \(3\) more apples, you now have \(4 + 3 = 7\) apples. Similarly, you have \(9\) apples and give me \(3\) of them, you now have \(9 - 3 = 6\) apples.
What if you have \(3\) apples and give me \(5\) of them, we have \(3-5 = ?\). This doesn't make sense you can't give me \(5\) of something you only have \(3\) of, but lets take the idea of subtraction, abstract and generalize it a bit, even if it doesn't make sense within the context of apples, and see if we can find something useful that can be applied to other things.
If you had \(3\) apples, and you wanted to give me \(5\), we can break down the \(5\) into two different 'transactions' where first you give me \(3\), and then you give me \(2\). So we can write this as \(3 - 3 - 2\), when you give me your \(3\) you have \(-2\) apples left. While you can't have \(-2\) of something, we can take the '\(-\)' sign to mean that you owe me \(2\) apples in the future.
This may seem like a lot, and a weird rule to associate owning negative something as owing in the future, but this is exactly how banks work! If you have \(\$50\) in your bank account, and you buy something with a debit card for \(\$60\), your account balance is now \(-\$10\) or you owe the bank \(\$10\). By generalizing and abstracting the concept of subtracting, even though it doesn't make sense to have \(-2\) apples, we were able to find a very useful application of this new way of thinking about subtraction.
Throughout the lessons, I'll give examples and applications on 'why' we are doing something.
With that in mind, this is the first section that will go over how we work with different types of numbers (fractions, decimals, negatives, etc...). The first part and primary focus will be on...
- understanding the different types of numbers
- understanding math notation (primarily set theory)
- adding and subtracting
- multiplying and dividing
- exponentiation and radicals
- solving equations by using the order of operations
This should be all you need to get to algebra where we will go over this again, but in a more generalized way where we will go to graphs applications.
After finishing the section on solving equations, I'll give some overviews and visuals of some cool stuff such as using imaginary and complex numbers, different counting systems, and some other weird stuff like quaternions (which are a 4 dimensional number that has tons of applications specifically when dealing with rotating things in 3d space).
With all that in mind, math is hard for everyone. In my experience, the hardest math class is the first one that you really take serious. Don't get discouraged!
Let's get started!