Although you may have plenty of occasions where you need to deal with numbers, many of you might feel inadequate in numbers and calculations.
It is certainly true that mathematics, which requires complicated formulas and functions, might be difficult to understand, and could be quite challenging to re-learn once you finish your education. However, in many cases just a simple arithmetic is enough to make use of, in a daily life as well as the world of business.
Dr. Brian Kernighan, a professor of computer science at Princton University and the author of “Millions, Billions, Zillions: Defending Yourself in a World of Too Many Numbers“, teaches us how to become familiar with arithmetic.
This interview is based on the themes of the following book:
“Millions, Billions, Zillions: Defending Yourself in a World of Too Many Numbers“
Japanese translated version is here.
Dr. Brian Kernighan
Professor at Princeton University
Brian Kernighan received a PhD in electrical engineering from Princeton in 1969. He joined the Computer Science department at Princeton in 2000, after many years at Bell Labs.
He is a co-creator of several programming languages, including AWK and AMPL, and of a number of tools for document preparation. He is the co-author of a dozen books and some technical papers, holds 5 patents, and is a member of the National Academy of Engineering and of the American Academy of Arts and Sciences.
His research areas include programming languages and software tools. He has also written several books on technology for non-technical audiences: “Understanding the Digital World” (2017), “Millions, Billions, Zillions: Defending Yourself in a World of Too Many Numbers” (2018) and “Unix: A History and a Memoir” (2019). His most recent book, “The AWK Programming Language, 2nd edtion”, was published in 2023.
Your arithmetic ability improves by repetition of even very simple calculation in your daily life
Sato: My first question for you is, what kind of thinking habits do people who excel in arithmetic develop?
Dr. Kernighan: Before answering your question, let me clarify the difference between arithmetic and mathematics. Math is more complicated and advanced, and it is much less likely that ordinary people are going to use it regularly. However, you certainly use simple arithmetic almost every single day in your daily life, at the level you would learn in American elementary schools.
In terms of habits, you might wonder whether you actually use some of the arithmetic that you learned in school long ago in day-to-day life. However, what I’ve been trying to explain here is that arithmetic is very useful to apply in adult life as well as in school.
Many of the numbers that you see in day-to-day life are presented by people who are trying to sell us something, to get our vote for someone, or convince us of something to get your support for them. You should approach them with skepticism, and should be very cautious about that. If somebody tells you a number, think for a moment about why that number is interesting or important or relevant, and how likely it is to be correct. Attention to the background things would be important.
Another aspect I would like to mention is that lots of the uses of arithmetic that I would be talking about refer to really simple arithmetic. Multiplication and division of approximate numbers or very rough computation is often enough for this purpose. You don’t even need a calculator or an app on your phone. Pencil and paper might be helpful but sometimes you don’t even need them.
What’s important is keeping things quite simple and straightforward, and applying arithmetic to crucial things in your life. The thinking process of questioning “why?” when you look at numbers or when you do some kind of calculations is the most important, for both in your personal life and in your life as a Japanese citizen.
Sato: Thank you very much. The next question is, what kind of studies should someone who is not good at arithmetic undertake to become proficient?
Dr. Kernighan: I think that’s like everything else in the world. The more you practice arithmetic, the better you are at doing it. However, for some things, it takes a longer time. I have never mastered Japanese, but I think I could have improved my Japanese skills if I had practiced more. The same goes to arithmetic. The more times you actually do it yourself and think about it, the more likely it is that it will get easier as time goes on. One of the things that I would like to suggest is to make estimates of things by yourself, things that are in your life that where you could do some estimation.
I’m giving you a random example. Suppose that I would like to go from Tokyo to Osaka. By shinkansen it will be around 3 hours. How long would it take you to walk? It is a little over 500 kilometers between Tokyo and Osaka and you can walk perhaps 5 to 6 kilometers an hour. Therefore, it will probably take you 100 hours of walking. You might think it is a silly example but using a nice round number like 500 and 5 is actually a good way to go. Dividing 500km by 5 km per hour is quite easy, no need for a formula nor a calculator.
Using numbers helps you make sense of the example you come up with and makes it easier for you to decide whether that value is valid or not by applying your common sense. Is taking 100 hours of walking from Tokyo to Osaka a reasonable estimate, even for people living in Tokyo and having a better common sense about the land they live in? You should note that my example assumes that you don’t even stop, sleep, nor eat, and continue to walk 5 days, 24 hours a day, so if you would like to be more precise, you will need some adjustments. And, of course, there will clearly be a difference depending on whether you follow the train tracks, or the road or some path along the coast.
This kind of practice in making estimates will improve your ability at arithmetic. As I said, rough computation with approximate numbers is good enough. One of the things that I found helpful is giving an estimation problem to my students and tell them to hold on and not go to Google directly because what the important thing is you use your brain and come up with some answers. Think it through without using the internet or the calculator, come up with your answer, and then discuss it with others to see whether you got close or whether you’re far away. And with that feedback you’ll improve your ability, and you will eventually be more confident about your arithmetic skill.
Use Law of 72 to gain an insight
Sato: Now I see that there are plenty of opportunities in our daily lives where you could make use of arithmetic, but many of us overlook them.
Dr. Kernighan: There are some shortcuts that you could actually learn that will make life simpler. One of them is called The Rule of 72, which tells you how long it takes something to double. For example, suppose the rate of inflation in Japan is 3%. How long from now would your money be worth half of what it is, or how long from now will something cost twice as much as it does today. The Rule of 72 says, take 72 and divide it by that given percentage. In this example, 72 divided by 3 equals 24, so in roughly 24 years, the prices will be doubled.
Another example might be the headlines I’ve seen lately, which is that the population in Japan is declining by roughly 1% a year. How many years will it be before Japan has 60 million people instead of 120 million? According to The Rule of 72, it’s going to be about 70 years because dividing 72 by 1 equals 72. So, 70 years from now, the population will be half of what it is today. There are obviously a lot of factors affecting the change in population, so this prediction is not absolute for sure. And yet The Rule of 72 only requires quite simple calculations and gives us the basic idea of what would happen in the future.
There’s another trick that tells you how much of the share you would get. For example, take the national debt or the budget in Japan and divide it by 100 million. That’s your share. And working backward could be helpful in order to detect errors. Suppose that I figured out that I could walk to Osaka in 10 hours, which would imply I walk 50 kilometers an hour. That is when you realize there’s something wrong with the earlier answer you came up with because you can’t walk 50 kilometers per hour.
All of this stuff is going to be useful, and it also could be fun if you get into the habit. If you start doing it even a bit, you will start to get better at it and you would feel more comfortable in your judgments. You also would be able to spot things that are wrong, and perhaps profit from that.
“Approximate” is good enough, doesn’t have to be precise numbers
Sato: Thank you. What point should one be mindful when interpreting statics or graphics, because there may be many occasions where you deal with them in business settings?
Dr. Kernighan: That’s more or less the first thing that I suggested here. Be very cautious, alert, and skeptical when you see statistics and graphs.
More specifically, one of the things is the difference between the average value and the median value. So, here’s an example. Who’s the person that everybody in Japan would recognize as extremely wealthy? Musk, Bezos, Gates, or somebody else, who would you pick?
Sato: Umm, for me it’s probably Gates.
Dr. Kernighan: Alright, let’s go with Gates. So, there are four of us here in this Zoom meeting. Make up a number of our average net worth. It could be ¥1,000,000 or ¥10,000,000. That’s the average value of four of us here. If Bill Gates were to join this conversation, the average value of net worth would skyrocket. It would be enormous because Bill Gates is worth probably around 100 billion US dollars. However, the median would not change if Bill Gates came in because the median value would be the middle value of this group.
The takeaway from this example is that when somebody says, “the average value is…”, then you need to be cautious. Because if the date includes people like Bill Gates, it will skew the whole thing and push things so far to the side that the value isn’t representative. For some things, average is fine, and for other things median is better way to capture the important part of a set of numbers. And all the examples I talked about doesn’t require any math. You just need to think, “are they giving me a median or an average?”.
Another example that I see a lot is excessive precision. For instance, the population of Japan today is about 125 million. Now, if somebody tells you that the population of Japan is 125345678, do you assume it is a correct number?
Sato: You are not able to know the population of Japan that accurately at any given moment because it’s constantly changing.
Dr. Kernighan: Exactly. That’s wrong because you can’t know it that accurately. If they just use a calculator, it would give them 10 significant figures, but most of which are meaningless noises. This excess precision is usually a sign that somebody was doing something wrong. That is, when somebody tells you that the population of Japan is 125345678, it is likely that they are presenting a fictional number with too much precision.
Sometimes people present those numbers because they are ignorant, and other times because giving a very precise number sounds more authoritative and right while it is not. Again, when you see excessive precision, you have to doubt it.
There’s an example in the book that I wrote some while ago. The United States is one of the 2 countries in the world that still uses the old English system of units. Converting between English units and metric units is something that people do with calculator or apps and get some numbers with excessive precision. For example, the height of Mount Fuji is about 3,700 meters. When you take that number and convert it using a calculator, you get 12,139ft 1.29156…, which is unnecessarily precise.
Watch out for confusion of units
Sato: As you pointed out, in many daily occasions, you only need approximate numbers, not precise ones.
Dr. Kernighan: Another place where people make mistakes that you have to be alert for is when they use the wrong units.
They use days when it really should have been weeks or vice versa, or they use months instead of years and vice versa. You have to be very careful about what the units are, whether they are correct and whether they are consistent within the sources. If you accidentally use months instead of years, that’s a factor of 12 potential error.
Another example of that, which shows up a lot in United States and probably other places, are errors when you’re talking about huge numbers like millions and billions and trillions. The meaning of the word “billion” is somewhat different in different parts of the world. In Japan, if somebody translates “billion”, it is 10th to the 9th power. But in a different place a billion means 10 to the 12th power. Getting that wrong would lead to a huge mistranslation and misunderstanding. I see examples of this almost every week in even reputable papers. If somebody were to increase your pay by a factor of 1000, you’ll notice it. It would change your life. If it went down instead, that you will change your life too. The factor of a thousand makes a lot of difference.
Another thing that you can watch for confusion of linear dimension, something like how long is the side of a piece of land or something like that as compared to the area. Let’s draw a circle at the center of Tokyo. If we make it 1 kilometer in radius so that will encompass a certain number of people. If you double the radius of that circle, it will be four times as much area, therefore 4 times as many people. However, confusion of linear dimension and area dimension leads you to think that double the radius means two times as many people. Occasionally the same thing goes with volume dimensions, where things go up as the cube rather than as a square or linear.
Pay attention to numerical values and shapes in graphs
Sato: I see. The knowledges you mentioned would be certainly necessary to improve arithmetic skills.
Dr. Kernighan: Please note that there’s a lot of tricks that people use with deceptive graph.
There’s a wonderful book called “How to Lie with Statistics”, which was written by Darrell Huff in the 1950s. For example, if you have a graph that’s supposed to show how something is changing over time, you can make it look like a huge change if you discard the values below the minimum one, so the graph doesn’t start at zero, it starts sort of near the lowest value.
Or, think about a linear value like the height. We line people up, we have their heights, and we can see who’s taller and who’s shorter. However, once you draw pictures of people where their height becomes proportional to the volume, then you could amplify the heights difference than what it actually is. Depicting something that’s intrinsically one dimensional using 2 or 3 dimensions gives a wrong impression visually.
Something else that you want to pay attention to is axes on a graph. You have to look carefully to make sure that what’s on horizontal and vertical axes, whether they are actually correct, and that they are uniform and there aren’t omitted parts and so on. You often see this deception in political campaigns where they skew or stretch time intervals to make the data look like it’s changing in a way that it doesn’t.
One of the bad things about modern technology is that we have ever more powerful tools to draw quite nice and fancy graphics. And consequently, we get graphs that are potentially ever more deceptive. Sadly, it’s a fruitful area for trying to deceive people.
It’s never too late to improve your arithmetic ability
Sato: Thank you. I’ve learned a lot. The final question for you would be, could you please share a message for readers who find an arithmetic challenging?
Dr. Kernighan: I think it goes back to what I was talking about earlier. If you can get into the habit of looking at things, the arithmetic is actually quite simple.
When you see some random numbers while taking a walk, or it pops up in your head while you are talking with someone, try doing arithmetic. Multiply it or divide it by approximate numbers. If you find it difficult, round the number or replace a huge number to a power of 10. What is important is that you attempt to question something and get some answers using your own brain. Ask questions like “How long would it take to the destination by car?” “How about when you take the bus?” and come up with answer. Almost all the cases, an approximate answer is more than enough. As you keep doing it, you will get better at it.
Also, I think it’s very helpful when you do arithmetic to know some basic facts. Knowing the population of Japan will enable you to make an estimation. If somebody says that the budget in Japan is, whatever some huge number of yen, you can then divide it by the population, and you get your share of the Japanese national budget. So knowing populations, knowing distances, knowing areas, knowing something about money gives you a basis of facts that you can use to make an estimates and helps you get better at doing it.
You come up with a number or two. It could be nice round numbers like 1,2,5, or 10. Multiply it or divide it by approximate numbers. Once you get into the habit, I think it becomes self-reinforcing because you will find it fun when you figured it out. It’s kind of fun to be able to do that. It eventually leads you to learn more and find ways to practice it.
( Interviewer: Naoto Sato )