Everything you know about math?

Question

In sociology today, we had an odd request from our teacher. She told us we had fifty minutes to write an essay about everything we know about math. (After five minutes, she stopped us and asked us why we obeyed. That's the sociology part.)

It was an interesting question, and I'm actually a little sorry we had to stop. How would you have formed this essay?

I started off with the number 1 and defined the addition property so I could count all the natural numbers. Then I defined the subtraction property so we could have all integers. By this point, she stopped us, but I would have continued on to how addition begets multiplication, and division is reverse-multiplication and gets us our fractions. Then I would have defined exponents and radicals to make the irrational numbers, and finally square roots of negative numbers to make complex numbers. After that, I figure I probably would have run out of time.

How would you have written your answer to this? (Summary, please!)

Ignore the sociology part. This question is not about that at all, but rather how you would have written the essay if you had to.

Answer 1

I would start out by laying out what ZFC set theory is and how it is the basis of most modern mathematics. From there, I would introduce the Peano postulates and explain how they construct the natural numbers. I would then define addition and subtraction. Next, the natural numbers need to be generalized into the integers. From there, we can extend our definition of addition and subtraction and introduce multiplication and division. Division leads to the construction of the rational numbers. The irrationals arise naturally from there. Next, I would introduce exponentiation as repeated multiplication. Exponentiation leads to the concepts of roots, which introduce imaginary components to our numbers and yield the complex numbers. Here, I need to introduce the concept of the limit to define exponentiation to an irrational power. Limits are dependent on elementary algebra and the algebraic properties of numbers, so I would introduce algebra in order to get limits working. I could develop this far more than I need to in order to define limits. This would include stuff like the binomial theorem and the fundamental theorem of algebra. Next, I would take a little detour into geometry and trigonometry, so I could define the trig functions. From there, I would introduce Euler's formula which leads to a definition of exponentiation that allows complex exponents. I could go on to hypercomplex numbers, but really, who uses them? Now, the fundamental arithmetic operations are defined. I would flesh out trig and geometry. We have mathematics as taught up to precalc at this point in time. We can now move on to bigger and better things.

I would continue by introducing the properties of numbers. You know... concepts like being prime, composite, abundant, deficient, etc etc etc. From there, I would introduce the various theorems of number theory, like the fundamental theorem of arithmetic. I could then introduce congruences and the theorems that go along with them, like Fermat's little theorem and such. This is all very interesting stuff, but it isn't very useful unless you work in cryptography. Let's continue on to another subject.

Analysis is the language of science. I already touched on it a bit earlier when I used limits to define exponentiation. I would start out with differentiation, integration, and methods thereof. From there, I would link the two together with the fundamental theorem of calculus. Next, I could introduce infinite series and the techniques and theorems for dealing with them. Next, I would introduce vectors, the properties thereof, their uses, and vector operations. Next, I could introduce partial derivatives. These naturally lead to multiple integrals. Next, I would define line integrals and surface integrals. Now, I can state Green's theorem. Next up is Stokes' theorem, which generalizes to the divergence theorem. From there, I could generalize the divergence theorem into a theorem that links integration and differentiation in all forms.

From here, I could go on to linear algebra and differential equations, but it really is too painful at the moment. Imagine that I covered this.

There, I've summarized most of the math I know. :)

Answer 2

Ok, the math part I would have done the same thing you did and perhaps gotten into Algebra a bit depending on time. However the Sociology part is another question. Blind Obedience? Because she is the teacher, you are trained since early childhood to do what an adult or in this case and adult substitute tell you to do, and not rebel. We are taught that rebelling equals punishment. You expected some sort of punishment if you did not do as she said so you did as she told you to do, even though you knew it was a strange request in a sociology class.

Answer 3

If I had to write a 5-minute essay, I think it'd go like that:

Firts I was introduced to the natural numbers, that is, the positive integers. Then, I got to know they are not sufficient to do all math we need, so came the rationals. How beautiful, they are countable! But they aren 't enough and I was introduced to the irrationals, so forming our complete real field.

And then I was introduced to the fascinating world of limiting processes. Uncountabilty, infinitude, abstract concepts that make you see the beauty of Math!. Functions, complex numbers, derivatives, integrals, so fascinating. And also facinating are the relationships between such concepts!

Well, my time is up.

Answer 4

I could never write a definitive essay on this subject. One element of knowledge in maths (math is for the USA) leads to another, as in other areas of life.

I would build mine around sections of maths that can be studied. Up to O level these were arithmetic, algebra, geometry and trigonometry.

From then on, the various sections start to merge, and for A level you had Pure Maths and Applied Maths which relates more to engineering.

I have also a lot of interest in mathematical puzzles and fallacies, and would have to include a section on this to illustrate why I have found the subject both fascinating and frustrating for such a long time.

What can be included has to be based on the length of essay you are aiming for. Anyone who can write everything he knows on a subject cannot know much.

Perhaps you see now why I am better at maths than at essay writing.

Answer 5

Okay that's way too detailed. Here's how i would have answered: First off, i say something about math that's really interesting, like a saying or a joke about it. Then i would define mathematics and say a lot about numbers and theories and solving problems. And since math is not only focused on numbers, i would say something about the ideas of the people (who were great thinkers) who used math in their social philosophies and the like. Then i would end up with it's applications in our everyday life and so on. There, if that's how i approached it, i wouldn't run out of time. I hope this helps.

Answer 6

I'm with you. My essay

"1+1=2, everything is a variable of that."

Answer 7

I would have done my best to look like I was deep in thought, and then written something like

"All I know is that it is deeply, profoundly evil, and that is why I love it."

elaborating no further, aside from citing unspecified national security issues that would keep me from going into greater depth.

The situation would be so absurd that I don't think that I could manage to take it or her seriously.

Answer 8

If you give a definition of the word math it will be more complete than to give all information on things that are considered math.

a definition could be:
A logical language to solve questions with numbers.

Something like that.

Answer 9

I know:
- Logic
- Set Theory
- Everything else that I know can be derived from the above.
- What I don't know can also be derived from the above - I just haven' gotten around to it yet.

End of essay.

Answer 10

Very funny. Now you want to justify, your blind obedience.
It's difficult at your age, how ever adults follow the same pattern too. That's troubling, plausible deniability.