Eugenia Cheng’s How to Bake Pi | Summary

Eugenia Cheng’s How to Bake Pi was a pleasure to read. In her book, she talks about what math is, the nature of math, and category theory specifically. This book is very tangible and easy to read. Throughout her exploration of mathematics, Cheng often uses the metaphor of baking which I found very helpful.

The beginning of the book starts broadly, answering the question, what is math. According the Cheng, “mathematics is anything that obeys the rules of logic, using the rules of logic.” She describes the two ways to do mathematics. The first is abstraction. This is where we find math in seemingly unmathematical ideas. The second is generalization – where we build on things we understand.

In addition, Cheng notes that math seeks to make things simpler by noticing things which are the same, ignoring the small details. When we problem solve in math, it is crucial to be able to organize the information we have in order of significance. We need to be able to take reality, apply abstraction and bring it back to reality again. In order to prove something is true we use the mathematical method and use logic instead of evidence. In this sense, Cheng states “math is a world where the means justifies the end.” We use logic to justify our conclusions even if we can’t actually come to that conclusion using numbers. Similarly, we need to study exceptions and general bodies of behavior. Cheng points out that mathematics allows us to do this in a sense that we do not have to imagine them at all in order to study them.

Next, Cheng talks about what motivates us to do mathematics. She states, “the best mathematical inventions are the ones that make internal sense and solve some existing problem.” Where solving an existing problem may be an external motivation, we need to continually ask the questions WHY? like small children and seek to be internally motivated. Cheng points out that we often accept truth on the grounds of higher authority. We assume that they did the homework, so why should we. But “one of them aims of mathematics is to do things from scratch.”  Again, we need to understand the process, and when we do “math is easy”. There is no mysterious missing steps because each step of logic is connected to a preceding one.

We have applied math and we have pure math. According to Cheng, “applied math could be thought of as the theory behind things in real life and pure math is the theory behind applied math.” Pure math is like building a complicated structure out of lego blocks (no wheel or windows). Category theory is given in pure math to group similar things in mathematics. We group things by studying the context they are in in order to learn more about them. So what allows two things to be in the same group? According to Cheng, “two things are the same if you can reverse the process to get from one to the other” (i.e. they are invertible). Category theory looks at the role a thing plays but it also looks at a role and see what plays it.  In this way, category theory illuminates math and aids in our understanding.

Finally, my favorite quote in Cheng’s book is the following: “Knowledge is power, but understanding is more powerful.” Mathematics is about logic and process, and understanding is key to its power.

The Usefulness of Understanding Math in Accounting

As a math major and accounting major, I sit in the perfect seat to reflect on the usefulness of understanding math in accounting. I have been asked the question “are math and accounting related?” and my answer is “absolutely”.

In a broad way, my mathematical training has helped me think through various accounting concepts and solve accounting problems. It has expanded my brain to be able problem solve efficiently by taking all the information I have and sorting through it to find the good stuff. I feel as though it has also helped me understand concepts. Sometimes in accounting a concept is explained with an equation. When we add, divide, or multiply two concepts or numbers and yield a third, math helps me understand what is actually going on; I don’t have to go through the work of memorizing the accounting definition because I understand the role that these operations have. For example, the following is an accounting equation:

Unit sales to attain the target profit = (Target profit + Fixed Expenses) / Contribution margin per unit

This is hard to memorize if you don’t understand the terms and if you don’t understand the role of division in this example. Target profit is equivalent to contribution margin per unit times the number of units minus the fixed expenses. When we add fixes expenses to this quantity we are simply left with our contribution margin per unit times the number of units, i.e.:

Unit sales to attain the target profit = contribution margin per unit* units /contribution margin per unit

Contribution margin per unit cancels and we are lefts with the number of units. That being said if we understand the algebra used here and we know the concept behind our accounting terms, we can break it down to the basics and understand what is happening in these equations.

Here is another example. In my intermediate accounting class, we just talked about how to calculate the future or present value of loans/investments, bonds, and annuities. Here is an example of a question we might see: You invest 12,000 at 4% interest which is compounded annually. You want to know the future value of the investment after 12 years.  If you are familiar with multiplication/math, you might computer 12,000*1.04*1.04*1.04*1.04(12 times total). Then you might realize that this is the same as 12,000(1.04)^12. In my accounting class, we learn that to find the future value, you set up the following equation and solve. 12,000(FVF)=FV where FV is the future value and FVF is the future value factor. The FVF can be found by using a chart where n=12 and i=4%. The values are already computed and given to us. Even though it may seem like using the chart takes fewer steps and less though, I don’t like it. Understanding what is actually happening when we invest 12,000 and collect 4% interest every year is very clear when we actually do the math and think through 1.04 and why we are multiplying. However in the FVF function, it isn’t really clear why we multiplied. How does one calculate FVF? What if I don’t have the chart handy? Similarly if we want to find the present value if we know the future value (lets say it is 12,000 and interest is 4% compounded annually), the math major might do as follows:

12,000=PV(1.04)^12

12,000/(1.04^12)=PV

Accountants would set it up as follows.

12,000(PVF)=PV

where PVF is equal to the present value factor. Even though in this case we have the end result instead of the initial information, the calculations are the same; multiplication is still used.

This example above is a simple one, but it gets more complicated in accounting. We talk about bonds and annuities. Annuities are where we either add or deduct money from an investment throughout the period of the investment. Sometimes we wait 10 years before we start taking money out. These types of problems make for great math problems! When we use future value factors, or present value factors I think it takes the math out of it. It makes more complicated; it makes it more about memorization than actually understanding the concept. In this way, understanding the math in accounting makes accounting seem more simple because I am able to break it down to the building blocks. I don’t need to spend the time wrapping my mind around the math involved because I know what calculations are being done. I can focus on the concepts.