My rating: 4 of 5 stars
A few times I’ve seen postings on Facebook where people are proud of the fact that they “got through another day without using math”. I’m amused but a little sad that they think math is unnecessary in day-to-day life. I wonder if they really didn’t use math or did it without thinking of it as math.Or is it true that since they don’t have a background in math they just ignore the problems in their lives where math could help Now, I confess I was an English major and ignored math and the sciences; but I’ve come to undertstand that more math would have been helpful.
Steven Strogatz shows us the basic concepts of numbers and math, building from the simple: Sesame Street characters counting fish, to the mind boggling: some infinities are larger than others.
We first learn about the power of numbers when we go from calling out “fish, fish, fish” for each fish we see to grouping them together in the abstract idea of “three fish”. Numbers are abstract ideas we use to stand in so we can easily measure and compare things. Once we build a set of relationship rules (addition, subtraction) we continue to develop methods of relationships. For example we build fractions as “ratios of integers – hence teir technical name, rational numbers.” (p 29). These rules continue to build upon one another and take us through algebra and geometry to calculus. As an example Strogatz demonstrates that adding “all the consecutive odd numbers, starting from 1: The sums above, remarkably, always turn out to be perfect squares” (p10).
My biggest takeaway from the book is that when you have a hammer, everything looks like a nail. You can only use the tools in your belt to solve the problems you encounter. And worse if you do use the tools in your belt you may get the wrong answer. Or worse yet; you may have the correct tool set but use them dishonestly to misdirect people – those people like me – who didn’t study enough math.
An example of that is statistics, where figures lie and liers figure. Most of us have at least a passing understanding of normal distributions (bell curves). They “can be proven to arise whenever a large number of mildly random effects of similar size, all acting independently, are added together. And many things are like that.” (p 178). Many, but not all. “[P]lenty of phenomena deviate from this pattern yet still manage to follow a pattern of their own.” (p 178). But we are more comfortable with the normal distributions and have the tools (the mean average) to work with them. In Power-law distributions the “modes, medians, and means do not agree because of the skewed, asymmetrical shapes of their L-curves. President Bush made use of this property when he stated that his 2003 tax cuts had saved families an average of $1,586 each. Though that is technically correct, he was conveniently referring to the mean rebate, a figure that averaged in the whopping rebates of hundreds of thousands of dollars received by the richest 0.1 percent of the population. The tail on the far right of the income distribution is known to follow a pwoer law, and in situations like this, the mean is a misleading statistic to use because it’s far from typical. Most families, in fat got less that %650. The median was a lot less than the mean.” (p. 180)
I’ve been intimidated by calculus but Strogatz does an effective job of making it approachable – you won’t learn calculus from the book but you’ll get a glimmer of understanding. If we want to find the area of a circle we start by fitting a square inside and calculate its area; then turn it into an 8 sided figure – like slices of a pizza – and calculating its area we get closer yet. And so on as the number of pie slices approaches infinity.
Strogatz wraps things up with the theory of infinite sets using the illustration of the Hilbert Hotel which is always full but there is always room for one more. I can’t do it justice here but he shows how the infinity of the real numbers between 0 and 1 is bigger than the infinity of whole numbers. Whaaaat?
Finally I became acquainted with the “recreational mathemusician” Vi Hart through this book. She is a video illustrator who does some marvelous work demonstrating mathematic concepts. Even if you don’t read this book (which you totally should), check out Vi Harts story of Wind and Mr. Ug; a couple of two dimensional beings who live on a transparent Möbius strip.