“Not everything that can be counted counts, and not everything that counts can be counted.” — Albert Einstein

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Numbers.

Limitless in every dimension but seemingly always limited in our appreciations.

Since prehistoric times, our ancestors have sought to describe quantities. At first, it began with simple cave illustrations: We perceived three buffalo, and so, we drew three buffalo. In correspondence, the notion of natural numbers—one, two, three, and so on—grew to their modern maturity as we discovered more about this subliminal art of counting.

Eventually, this keenness to quantity spawned several symbolic representations. The Ancient Babylonians, Egyptians, Greeks, and Romans each took it upon themselves to demonstrate proper counting methodology (i.e. to illustrate numbers in universally recognized fashions). Consequently, dots, lines, fingers, swirls, crowns, and even frogs were used to represent the “counting” numbers. Given time for proper evolution, these numbers eventually became the familiar figures we use today—1, 2, 3, etc.

Of course, numbers extend far beyond the idea of “countability.” In other words, there are some numbers out there for which we simply do not say, “There are ‘x’ number of students in the classroom,” or, “That tree has ‘y’ number of leaves.” These numbers could be, for example, -3, pi, and ½. To illustrate this point more fully, I call upon the famed Greek mathematician Pythagoras, a man who struggled with the notion of what we now call irrational numbers.

To be a touch more precise than previously stated, Pythagoras went through several internal battles because of the number *sqrt(2)*, an irrational number very important to trigonometry. Using modern technology, we can approximate the number *sqrt(2)* fairly easily. (It is roughly equal to 1.41421356237.) However, despite how powerful our technology may be, even we cannot fully obtain the value *sqrt(2)* because its decimal component never ends (i.e. an infinite amount of numbers exist after the 7 in 1.41421356237). This concept troubled Pythagoras greatly because he could clearly visualize what *sqrt(2)* equaled geometrically using a theorem which bears his namesake. (See the diagonal line in the diagram.)

So, having seen *sqrt(2)* drawn as a line segment, I now present to you the very same question Pythagoras asked himself countless numbers of times: How is it that a number which theoretically never ends fits into a perfectly finite geometric observation?

The answer lies within the beautiful, and careful, construction of our number system. To read more about how we organize numbers (and reach a more satisfying conclusion to my question), click here.

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I finally read your blog! And I find it very intriguing.

Usually we are fascinated by solid and physical occurrences in nature. Math however, seems to allow a fascination of the ungraspable. Mathematicians derive complex formulas from other complex formulas, and many of those equations are not applicable to the physical world. It might seem at times as if math is a man made design, making it curious to think that we can be so captivated by a theory purely of the mind.

However! Your blog does a wonderful job of nullifying the above statements. By describing math starting with its early beginnings, you have illuminated the clear and concrete connection math has to nature. After reading your blog, I am confident that math has always existed, and will always exist as a way to describe the physical world, and that as we learn more and more about math, we are merely digging deeper and deeper into a preexisting explication of the universe.

Your passion for mathematics is fantastic! For me, math courses have always bestowed an abundance of distress, but I think your passion blogs can assist in mollifying my fears! I’m certainly looking forward to your future posts.

I also love that Einstein quote; it’s very clever.

I really appreciate your feedback, and I’m glad you enjoyed the first blog. Honestly, I wasn’t sure if people would want to read about math outside the classroom, so the fact that you even made it through that article is incredible.

Math truly gets a bad reputation for being repetitive, emotionless, and void of any meaning to the real world, but nothing can be further from the truth. Math is an art form (as you both got the impression), a way of critical thinking where, given some initial conditions, you can go as far as your imagination takes you. That sort of elegance and purity in design renders emotions that harken back to our very origins. It’s tradition. We can only expand our math knowledge with each coming generation–something which makes the subject special.

I’m not forcing you to fall in love with the subject. I don’t want to make that my intention. But, if I can change two individuals’ perspectives of math from dull to wildly imaginative, it would definitely mean the world to me.

Thanks again for your comments! I really appreciate them! 🙂

This is a truly fascinating blog. This blog gives a very counter-intuitive, yet important, view of mathematics, as math can be all too often dismissed as simple counting. I personally like this blog because it embraces the history and functionality of mathematics in a new, fresh way. This is how math should be thought of.

Well let me begin with saying “wow.” I am very impressed with your mathematical knowledge as well as your passion for it as an art. I suppose I never really thought about mathematical concepts as “beautiful,” but I can appreciate this opinion nonetheless. Math has always been an integral part of education; it’s quite interesting that you see it as so much more.