How Not to Be Wrong Book Summary

Critical and logical thinking is so necessary for success in this world. With the importance of logical thinking, why does it seem like so many people don’t know how to think? Wouldn’t you think you should learn how not to be wrong?

How Not to Be Wrong: The Power of Mathematical Thinking, by Jordan Ellenberg, is a blueprint for mathematical and logical thinking.

When going through school, the different theories and concepts we learn are not always presented with the everyday reasoning behind them.

How Not to Be Wrong gives you a framework for how to think logically, and presents it with everyday language and examples.

The best part? How Not to Be Wrong is non-technical, and no prior math experience is needed.

These math concepts are important for math purposes, and we can take these principles and apply them in our everyday lives for success.

The rest of this post includes a summary of How Not to Be Wrong, takeaways from How Not to Be Wrong, and a reading recommendation for you.

Basic Rule of Mathematical Life: If the universe hands you a hard problem, try to solve an easier one instead, and hope the simple version is close enough to the original problem that the universe doesn’t object.

Book Summary of How Not to Be Wrong

How Not to Be Wrong: The Power of Mathematical Thinking is an easy and enjoyable read about mathematical thinking.

How Not to Be Wrong is split up into discussing 5 different mathematical concepts:

  • linearity
  • inference
  • expectation
  • regression
  • existence

Ellenberg looks to explain each of these concepts using stories from the past and non-technical examples.

To read this book, you do not need any formal mathematical training – there are no equations and many examples use illustrations to help the reader visualize the material.

Below is a description of each section:


The first mathematical concept discussed in How Not to Be Wrong is linearity.

Many mathematical models are linear and are used incorrectly.

The point of this section is to make the reader aware that non-linearity, i.e. curved lines, are much more common in the real world.

Some topics Jordan Ellenberg looks at in this section include, but are not limited to, government fiscal policy (what tax rate maximizes tax revenue?), curves are straight locally but curved globally, and projecting obesity in 2048 (100% of Americans will be obese based on the current trend!).

Use caution when using linear regression.


The first mathematical concept discussed in How Not to Be Wrong is inference.

Many mathematicians use hypothesis testing to make conclusions about statistical tests. This is commonly taught in entry level statistics courses.

Ellenberg discusses how many people produced statistical “tests” showing that various messages were could be found in the Torah if you mix and match the letters in a mathematical way (Ellenberg says it is kind of bogus).

Ellenberg also discusses the hypothesis testing, p-values, and how to appropriately draw conclusions (essentially proof by contradiction).

Improbable things happen a lot – “what is improbable is probable.”


The first mathematical concept discussed in How Not to Be Wrong is expectation.

Expectation is the average value of a situation (put in layman’s terms).

In this section, Ellenberg talks about how multiple groups won large sums of money playing a lottery game in Massachusetts, and looks at common misconceptions regarding the aggregation of the average value of various situations (the concept of additivity).


The first mathematical concept discussed in How Not to Be Wrong is regression.

Mathematicians use regression and correlation analyses to to draw conclusions about two variables.

A common mistake is drawing the conclusion that variables have a relationship based on the data where it could just be chance they are related.

In addition, you have to be careful when assigning the direction of the relationship. In the 1950’s, there were multiple studies on smoking and lung cancer. A question a mathematician could ask is “Does lung disease cause smoking?” (this is probably not the smartest question to ask, but the data could suggest this relationship).

Correlation is not transitive (no correlation does not imply no relationship)


The first mathematical concept discussed in How Not to Be Wrong is existence.

This section talks about a few different situations where unexpected results arise. One of the situations is in a three person political race and using different ways to tally the votes will result in different outcomes.

You need to know that it is possible to create the scenario you are analyzing, and not coming up with an imaginary outcome.

It is very difficult to apply traditional methods to “unknown unknowns”.

Takeaways from How Not to Be Wrong

With every book you read, it is a must to have takeaways and actionable items to implement in life.

From How Not to Be Wrong, the main takeaway is when working with data, you need to be careful with your assumptions and analysis.

Thinking mathematically is all about being skeptical about your results, making sure you have the right assumptions, and doing your due diligence.

You can twist and lie with data very easily. It takes a disciplined person with the right training to be able to do what is right with your analysis, and be aware of when others are trying to skew the results.

By understanding the 5 mathematical concepts above, you will be able to navigate tough situations more easily and logically.

Genius is a thing that happens, not a kind of person.

Our Recommendation for How Not to Be Wrong

How Not to Be Wrong: The Power of Mathematical Thinking is an easy and enjoyable read, and will be influential to any reader.

Whether you are an analyst at a company working with numbers, someone who plays games of chance, or someone who wants to increase their intuition of the subject, you’ll definitely want to check out How Not to Be Wrong.

Readers: do you try to gain intuition about technical subjects? Do you think learning about math and logic is valuable in your life?