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The Building Blocks of Mathematics and their Beauty

Somewhere in a discussion here (I think it was on the "Thinking like a Theorist: Number Systems") someone said that all of mathematics can be built up from 1, -1, addition, and multiplication. I found this statement to be fundamentally correct (in fact, multiplication can be removed as it is repeated addition) and rather intellectually exciting and decided to explore it a bit. Clearly from this concept we can produce the integers, and from addition we can produce multiplication. We can produce the rational numbers if we conceptualize two integers and what factor one must be multiplied by in order to produce the other. Through Euclid's proof of the existence of irrationals we can produce them, and so on. The list is endless. The point is, it's amazing how fundamentally simple concepts have produced all of...this, the beautiful world of mathematics that we delve into every time we visit Brilliant.

Any thoughts on this from more experienced mathematicians and theorists?

Note by Tanishq Aggarwal
3 years, 6 months ago

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This is an extremely good train of thought, and one that has spawned entire fields of mathematics. There are some questions which you seem to hint at, which I'd explicitly state for others to think about.

  1. When we are first introduced to addition and multiplication, we can only add finitely many of them together. What does an infinite summation (or multiplication) represent? Is it 'legal'? Does it obey the commutative and distributive laws?

  2. What is the concept of infinity? Is it simply "bigger than any number"? Are there different concepts of infinity? How does infinity affect addition and multiplication? What is \( \frac{ \infty} { \infty} \)? Which is 'bigger', \( \infty + \infty\) or \( \infty \times \infty \)?

  3. What is addition and multiplication? Are there other ways that we can 'add' and 'multiply' in other systems (say colors)? What is the essential characteristics / qualities of addition and multiplication?

  4. Is there mathematics that no system can handle? Can we prove or disprove every statement?

  5. Is there an 'innate' idea of mathematics? Certain concepts which we take as obvious truths actually took a long time to develop. The concept of 0 was only introduced in 9th century AD by India. Using symbols to represent equations (i.e. algebra in high school) only developed in 16-17th century.

  6. Is there math that an isolated civilization will never discover because of the way that they decide to interpret the world? How different is the world if we started off learning about complex numbers immediately?

Calvin Lin Staff · 3 years, 6 months ago

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@Calvin Lin I know very well that infinity is not a complex number; it is, for the most part, an undefined concept. But if we assume that we use our operations like we do on complex numbers, it is easy to show that \(n^2>2n\), so we could very well start an entire branch of mathematics based on infinity if we begin to treat it like a complex number (which is currently abhorred elsewhere in all of mathematics). Tanishq Aggarwal · 3 years, 6 months ago

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@Tanishq Aggarwal Could you please elaborate on your claim? I'm not sure how you came up with the whole \(n^2>2n\). Also, I do believe that there is such a thing as complex infinity. Bob Krueger · 3 years, 6 months ago

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@Bob Krueger Ah. The inequality is a direct result from the last question in question #2. It can be proved for real numbers greater than or equal to 2 as follows: Consider \(n \geq 2\). We subtract 1 from both sides to get \(n-1 \geq 1\), and then square both sides to get \(n^2-2n+1 \geq 1\), from which follows the desired result. We know we can square both sides as both sides are clearly positive. Tanishq Aggarwal · 3 years, 6 months ago

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@Tanishq Aggarwal Thanks! I just wasn't sure what the inequality had to do with infinity. Bob Krueger · 3 years, 6 months ago

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@Tanishq Aggarwal The above clam can also be easily proven to hold for negative real numbers, as the square of a negative is always positive, which is greater than a negative. Tanishq Aggarwal · 3 years, 6 months ago

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@Calvin Lin I'd like to point out that the concept of 0 innately existed long before it was mathematically developed. I think it is universal across all organisms of the concept of having nothing, due to the sense of homeostasis and knowing what perfect balance is within an organism's self and its environment. Additionally, many remote civilizations, before they were discovered, had no concept of numbers either. I'd like to point you to the Piraha tribe in the Amazon (I only know of them due to a recent Smithsonian documentary I saw), which have no number system, and see 1 as very little, 3 as few, 5 as a bit more, and 10 as a lot. Tanishq Aggarwal · 3 years, 6 months ago

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@Tanishq Aggarwal Very true. I believe algebra also existed innately, before it was truly invented. If a wall takes twenty bricks to make, and I have ten, how many more bricks do I need to make the wall? I don't need algebra to figure out the answer, I just subtract the two. I don't worry about doing to one side as I did to the other. Yet, these questions still reflect algebraic principles. Bob Krueger · 3 years, 6 months ago

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@Calvin Lin I have created a new discussion based on these questions. My intent was not to overshadow this thread, but to increase the longevity of this discussion. I think this more concise form will allow more users to respond more easily than in this thread, and that, of course, the discussion will last longer. Bob Krueger · 3 years, 6 months ago

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@Calvin Lin Zeno's paradoxes caused the creation of thought behind the infinite summation. The Greeks were, undeniably, stumped! The concept of the limit solved this problem by showing that if the length of the summation grows, the precision of the number it defines will increase, up to the point that if we have a series (infinite summation), the number we are trying to be define will be obtained.

Without this limit concept, we can't possibly make the link between the decimal system and fractions. A non-terminating decimal describes some number with a sum of infinitely smaller numbers. We can't define 0.33333.... as 1/3 without it. Tanishq Aggarwal · 3 years, 6 months ago

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@Tanishq Aggarwal should 0.333333.. (recurring) even be equal to 1/3... can you prove it satisfactorily? supposedly it depends on which exact number theory you decide to go with... Ben Blayney · 3 years, 6 months ago

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@Ben Blayney This is very easy to prove. Let \(x=0.333333...\) so that \(10x=3.333333...\). Subtracting x from 10x we obtain \(9x=3\), from which \(x=1/3\) is a simple result. Tanishq Aggarwal · 3 years, 6 months ago

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@Tanishq Aggarwal Notice that this proof rests on the assumption that \(\infty +1=\infty\). Bob Krueger · 3 years, 6 months ago

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This is the beauty of the most well known system in mathematics. I think that it is the sole drive of mathematics to produce more out of a few initial constraints. No matter what field you look in, it always starts off with a few rules (axioms) and new things are derived (theorems).

Back to the numbers, there are some problems with that initial system. For example, defining multiplication by using addition makes the definition only valid for integer values, as it would yet not make sense to add a fraction of some number. Other problems are with the transcendental numbers, posted in the original comment, which can not be defined with finite multiplication and division.

Mathematics is more than just creating these situations and following them to fruition. It's also about patching up the holes so create a more cohesive system. By the way, the person you speak of is me. :)

Original Comment Bob Krueger · 3 years, 6 months ago

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@Bob Krueger Could someone please explain to me why this is a bad comment? Bob Krueger · 3 years, 6 months ago

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@Bob Krueger I think people disagree with the fact that defining multiplication using addition makes the definition only valid for integer values. As I stated above, we defined multiplication for non integral values by considering two integers \(k\) and \(m\), with \(k\) not divisible by \(m\) and considering the value that, when multiplied by \(k\) produces \(m\). And the transcendental numbers \(e\) and \(\pi\) can be defined with finite multiplication and addition, as someone else pointed out in "Thinking like a Theorist" by defining \[e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}...\] \[\pi=\sqrt{6(\frac{1}{1}+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}...)}\]

I think people also disagree with your last statement because by the way it is built (a set of simple axioms and then a buildup upon them) there are incredibly few "holes" as you say. The few holes that have ever been found were things like Zeno's paradoxes and Russell's paradox. Tanishq Aggarwal · 3 years, 6 months ago

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@Tanishq Aggarwal Thank you. I appreciate it. I just find it interesting how you can feel like starting out with a certain set of rules will get you to what you want to prove, but sometimes you won't quite make it. The idea was that maybe we could only assume 1, -1, and addition to create all the numbers, but it turns out we can't make these transcendental numbers out of a finite sequence of operations. Therefore there was a hole in our original set of axioms, something that we as mathematicians have to recognize and patch up as seemlessly as possible. Those are the kinds of holes (in our reasoning and assumptions) to which I referencing. :) Bob Krueger · 3 years, 6 months ago

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@Bob Krueger We have defined limits, so that as the length of the series increases the precision of the transcendental increases as well. In fact both of these stem directly from Taylor series of trigonometric functions, and Taylor series are based on the derivative which is based on the concept of the limit. And limits are more of an intuitive addition to mathematics. So maybe they must be included in our fundamental building blocks of mathematics? Tanishq Aggarwal · 3 years, 6 months ago

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@Tanishq Aggarwal Interesting. So if we can assume that limits on polynomial functions makes intuitive sense, we can construct some transcendental numbers. But will this work for all transcendental numbers? I conjecture that it won't. Also, you falliciously said that those infinite sums of e and pi are finite multiplication and division. But really, if we assume the concept of a limit, we can create such numbers in that fashion. Bob Krueger · 3 years, 6 months ago

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@Tanishq Aggarwal Just pointing it out... There are slight errors in both your equations. See my original comment here. Mursalin Habib · 3 years, 6 months ago

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@Mursalin Habib Oops...thank you for pointing that out. I fixed the error. Tanishq Aggarwal · 3 years, 6 months ago

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I AM VERY INTERESTED IN THE CONCEPT OF INFINITY..Can anyone tell me all existed properties of infinity.? Friend Karl · 3 years, 6 months ago

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@Friend Karl There are many properties, for example, we define the absolute value of \(\frac{1}{0}\) as infinity. However, infinity is a concept our minds are seemingly only able to grasp when things become infinitely smaller, not larger, because dealing with infinitely large quantities is beyond the scope of our imagination. I think the next huge advancement in pure mathematics will be a rigorously interconnected branch of studies behind infinity and its properties--currently our closest reach towards this is the limit in calculus. Tanishq Aggarwal · 3 years, 6 months ago

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@Tanishq Aggarwal Don't forget Cantor's set theory and the different levels of infinity. It may not be complete or the most rigorous thing there is, but it gives even more knowledge about infinity. Bob Krueger · 3 years, 6 months ago

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@Bob Krueger Good point. Tanishq Aggarwal · 3 years, 6 months ago

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