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  • Writer's pictureRich Dlin

Imagine a World Without Algebra

Can you imagine a world without algebra?


I know what you're thinking. Yes, you can. You may even be thinking that it would be a better world, depending on your experience with algebra, I suppose. But that aspect notwithstanding, there is a high probability that you think algebra is an incidental tool, at best, and not something that anything particularly momentous hinges on.


It's not true. But I can see why people might think that. It's because algebra is widely and deeply misunderstood. And no, I'm not talking about grades in math. I'm talking about what algebra actually is.


So I'll do my best to explain it. Let's start with an example.


Suppose you are designing a metal cylindrical can that will be used for a new brand of energy drink. The aluminum that is used for the side and bottom of the can costs $0.03 per square centimeter, while the metal that is used for the top must be stronger, so it is a mix of aluminum and magnesium, and so it costs more - say $0.07 per square centimeter. Furthermore, the can you want to design will need to hold one liter of this drink. Now I get that at this point this sound suspiciously like a word problem from a high school math course, but I assure you, it is not. It's just a typical project that manufacturers around the world deal with regularly. So here's the kicker: there are lots of different possible dimensions for the can. It can be very tall and narrow. or short and wide, or something in between. The question is, what are the dimensions that will minimize the material costs to produce this can? Or does it even matter? Maybe, as long as the can holds the liter of drink, the material cost is always the same.


Well, it's not always the same. And there is a way to determine the dimensions that will minimize the cost. What you do is you guess at the dimensions, making sure that they will give the desired volume. Write those dimensions down. Then you manufacture a can with those dimensions, and weigh it (weigh the top separately), so that you can calculate the material cost. Write down that cost. Then, guess new dimensions, manufacture, weigh, calculate, write down that cost. Then, guess new dimensions, manufacture, weigh, calculate, write down that cost. Keep going until you have guessed all the possible dimensions. Find the lowest cost you wrote down. Bingo. You have maybe found the dimensions that minimize cost.


Or ... you could trust a mathematician, and find our that the optimal dimensions can be found using the following steps:


First, the optimal height and radius will always be in the ratio (A + B) : A, where A is the unit price for the side and bottom, and B is the unit price for the top. This is independent of the volume of the can. This means that if the volume we want is V, the radius we want will be


Once you plug that into a calculator, using the values A = 0.03, B = 0.07, and V = 1000 (because 1 liter is 1000 cubic centimeters), you will have the optimal radius, which is about 4.57 centimeters. Since the ratio of height to radius is (A + B) : A, which in this case works out to 10 : 3, the height is 10 times the radius, divided by 3, which will work out to about 15.24 cm.


It would probably have been hard to guess those dimensions.


There are really only a couple of things I want you to notice about the steps. First, note that the neither the formula for the ratio of the height to the radius, nor the subsequent formula for the optimal radius presume to know the prices or volume ahead of time. This means that if prices change, or if we are in a situation where the can we are designing is for a different volume of liquid, or using different materials with different costs, the formulas are totally reusable. This is called generalization, and it is one of the main powers behind algebra. Second, there is no explanation given for why those formulas work. And none is needed, if you trust the mathematician that developed them. However, if you wanted an explanation, you would see that it is more algebra, as well as some calculus, that is used to deduce the formula. That work is arguable difficult (depending on your expertise in math), but only needs to be done once, because everything is generalized. That means that with the "hard" work done, we now have a reusable formula that is flexible enough to be used in many situations. That's what algebra does, and if you start to think about it, you will see that it is used constantly, in many every day situations - not just manufacturing aluminum cans.


You might find it interesting to know that we didn't always have this tool. In fact, it was in the 9th century when Muhammad ibn Musa al-Khwarizmi literally wrote the book on algebra. The title of this book, translated in English, is The Compendious Book on Calculation by Completion and Balancing, or transliterated from Arabic: Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala. Trivia fans will rejoice in the fact that if you look closely at the Arabic title, you can see the origin of the word "algebra" right there, and in fact the book is often simply referred to as "Al-jabr". Those same trivia fans might also get a kick out of the fact that the word "algorithm" originated from al-Khwarizmi's name. In this book what al-Khwarizmi did that had never been done before is to use symbolic placeholders instead of actual numeric values, then write out algorithms using those placeholders that then could be used to solve common problems. If you look back at the example above, you will see that's exactly what happened. The symbols A, B, and V were used in place of numbers, and then an algorithm for calculating radius and height was described in this line:


But that algorithm is expressed using mathematical notation, which is very concise. It could also have said:


"First, multiply the desired volume by the unit price of the side and bottom of the can, then divide this result by the product of pi and the result of the sum of the unit price of the side and bottom and the unit price of the top. Finally, determine the cube root of this result, and you will have the optimal radius."


Which is pretty wordy, but accurate, and also algebra! In fact that paragraph is more along the lines of the algebra of al-Khwarizmi than the more mathematical looking expression you see above it. Using letters and mathematical notation instead of words is something that actually came much later, when near the end of the 16th century a French mathematician by the name of François Viète developed the concept, which was then heavily popularized by the work of René Descartes. Their work greatly simplified the use of algebra, and also led to what is known as modern algebra, which is what we learn in school. I find that very cool, personally, but I also understand that what it does is make algebra seem like something very remotely removed from the "real world". But it is not, and now that you have read this far, perhaps you will see how each of the following real world activities is actually algebra in use (I will use boldface font to highlight the variables):

  • Max is barbecuing ribs for his extended family. He usually makes 3 racks, but today he needs to make 7. Since 7 divided by 3 is about 2.3, Patrick takes the sauce recipe he normally uses and multiplies all the quantities by 2.3 to ensure his sauce will taste the same as it always does.

  • Barbara is cutting vertical blinds to fit into a triangular window. Barbara wants the slats to be 6 cm apart. She cuts one slat to fit the height of the tallest part of the window. Then she takes that height and divides it by the horizontal distance from the tallest part of the window to the bottom corner of the window, and multiplies that result by 6 to determine how many centimeters shorter each successive slat has to be.

  • Ray opened a tax-free savings account (TFSA) with an initial investment of $500. She also invested $700 in a mutual fund at the same time. A year later, Ray's received a statement giving her the balance in the TFSA, and 3 months after that she received a statement with the value of the mutual fund. To determine the annual interest rate she had yielded in the TFSA, Ray took the balance there, divided by $500 and subtracted 1. Then she took the balance in the mutual fund and divided by $700, but to account for the extra 3 months (for 15 months instead of 12), she raised that result to the exponent 12 / 15 = 0.8, before subtracting 1 to determine the annual interest rate she had yielded in the mutual fund. Ray then used these rates to decide if she wanted to shift any of her investments into the other account.

Note that in the list above there is no explanation as to why these algorithms do the job they are meant to do, although the explanations themselves are fun. However in each case there is a step or series of steps that can be followed using actual numbers in place of the unknowns, which will result in correct values to make decisions with. It's hard to imagine a world where we did not have that power! The power of algebra.


Thanks for reading,

Rich

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