Lets Try and do a ‘Workout’ with Math
Professors will tell you that Math uses many made-up rules to create relationships and models. When learning these rules you have to ask:
- What relationship does this model represent?
- What others share this relationship?
- Does the relationships make sense?
These are simple questions to ask, however, they help understand maths as a topics. Many people have struggles with math and need resources to helped them.
Textbooks rarely focus on easy understanding, we hope our blog will help in solving the ‘understanding’ of the math problems and formulas you are trying to grasp.
- The Pythagorean Theoreis not just about triangles. It is about the relationship between similar shapes, the distance between any set of numbers, and much more.
- e is the fundamental relationships between all growth rates.
- The natural log is about the amount of time things need to grow.
We want you to share your insights with us. The more we share together, the more we understand, the less pain we all have in achieving our goals, and everyone wins.
Math Evolves Over Time
Some say math is a way of thinking, and it’s important to see how that thinking has developed so we can get a better understanding of the changes.
Imagine you’re a pioneer doing math for the first time. One of the first problems will be how to count things like the number of corn cobs you have harvested. Several systems have developed over time to do this:
It was widely understood that no system is right for all applications, and each has advantages:
- Unary system: Lines of match sticks is a good analogy. It is as simple as it gets. Great for keeping score in games and you can add to a number without erasing and rewriting any formula.
- Roman Numerals: More advanced unary, with shortcuts for large numbers.
- Decimals: Huge realization that numbers can use a “positional” system with place and zero.
- Binary: Simplest positional system (two digits, on vs off) so it’s great for mechanical devices.
- Scientific Notation: Extremely compact, can easily gauge a number’s size and precision (1E3 vs 1.000E3).
It has been estimated that in 1000 years we will have a system that makes decimal numbers look as quaint as Roman Numerals.
Let’s think a bit more about numbers. The example above shows our number system is one of many ways to solve the “counting” problem.
The Romans would consider zero and fractions strange, but it doesn’t mean “nothingness” and “part to whole” aren’t useful concepts. But see how each system incorporated new ideas.
Fractions (1/3), decimals (.234), and complex numbers (3 + 4i) are ways to express new relationships. They may not make sense right now, just like zero was hard to understand by the Romans. We need new real-world relationships for them to be fully understood.
Even then, negative numbers may not exist in the way we think, as exampled here:
Negative numbers are a great idea, but don’t inherently exist. It’s a label we apply to a concept.
Some say they do.
Ok, lets have them show me -3 corn cobs.
Well, assume you’re a farmer, and you lost 3 corn cobs.
Ok, you now have no corn.
I mean, you gave 3 corn cobs to a friend.
He now has 3 corn cobs and you have zero.
The arrangement is that he owes you them.
So the actual numbers look like this (-3 or 0) depends on whether he’ll pay me back. I didn’t realize the opinion changed how counting worked. In anyone’s world, I had zero the whole time.
In reality it’s not like that. When he gives you the corn cobs back, you go from -3 to 3.
He does return the 3 corn cobs and we jump 6, from -3 to 3…
Confused, welcome to math!
Negative numbers can express a relationship:
- Positive numbers represent a surplus of corn cobs
- Zero represents no corn cobs
- Negative numbers represent a deficit of cows that are assumed to be paid back
But the negative number “isn’t really there” There’s only the relationship they represent (a surplus/deficit of corn cobs). We’ve created a “negative number” model to help with bookkeeping, even though you can’t hold -3 corn cobs in your hand. This interpretation of what “negative” means is a different counting system, just like Roman numerals and decimals are different counting systems.
Negative numbers are not accepted by many people, including Western mathematicians, until the 1700s. The idea of a negative was considered “absurd” than. Negative numbers do seem strange unless you can see how they represent complex real-world relationships.
Why Start Math Gym with Philosophy?
Deep understanding of the origins is the key to learning. It helps you arrive at deep insights, specifically:
- Factual knowledge is not understanding. Knowing “Screw drivers screw screws” is not the same as the insight that any hard object (a stone, a pan) can hit a nail home.
- Keep an open mind. Develop your intuition by allowing yourself to be a beginner again.
Call back to see our ‘math workout’ evolve…help us help others to grasp math by becoming a contributor. We would love to hear from you and publish your contribution. Reach out to us via the contact us page.
We acknowledge the sponsorship of Math Gym by Dux College who provide HSC Maths Tutoring as well as tutoring in other subjects in Sydney.