Book review of The Equation That Couldn’t Be Solved by Mario Livio

The Equation That Couldn't Be Solved


“The Equation That Couldn’t Be Solved” by Mario Livio delves into the history and significance of the quintic equation—a polynomial equation of degree five—and the discovery that not all such equations can be solved with a straightforward algebraic solution. This realization marked a pivotal moment in mathematics, leading to the development of group theory and further innovations in the field.

Key Themes

The book explores several key themes, including the nature of mathematical discovery, the evolution of algebra, and the impact of group theory on mathematics and beyond. Livio uses the stories of Évariste Galois and Niels Henrik Abel, two mathematicians who contributed significantly to the theory, to humanize these mathematical concepts and highlight the drama of their discoveries.

Examples Used!

Livio provides numerous examples to illustrate complex mathematical ideas in an accessible manner. He discusses how the solutions to quadratic, cubic, and quartic equations were found, setting the stage for the problem of the quintic equation. The narrative then delves into group theory, explaining how this branch of mathematics provides a framework for understanding symmetries in nature and in mathematical objects.

The following table outlines the key techniques and examples used by the author.

Chapter Title Key Concepts & Examples Including Appendix Problems
1 Symmetry Introduction to symmetry in mathematics and nature.
2 eyE s’dniM eht ni yrtemmyS Exploring symmetry from a psychological and perceptual perspective.
3 Never Forget This in the Midst of Your Equations The importance of symmetry in mathematical equations.
4 The Poverty-Stricken Mathematician Life of Niels Henrik Abel and his contributions.
5 The Romantic Mathematician Life of Évariste Galois and his contributions to group theory.
6 Groups Introduction to group theory and its foundations, including the Galois Family Tree from Appendix 8.
7 Symmetry Rules The application of group theory in understanding mathematical and natural symmetries.
8 Who’s the Most Symmetrical of Them All? Examining objects and concepts that exhibit high levels of symmetry, featuring the Solution to the Matches Problem (Appendix 10).
9 Requiem for a Romantic Genius Reflecting on the legacies of Abel and Galois, with additional context on Adriaan van Roomen’s Challenge (Appendix 6).


The book concludes with a reflection on the beauty and universality of mathematics, as revealed through the story of the quintic equation and group theory. Livio argues that these mathematical concepts not only advance our understanding of the mathematical world but also have practical applications in various fields, from physics to cryptography. He emphasizes the importance of curiosity and persistence in the pursuit of knowledge, celebrating the contributions of Galois, Abel, and others to our collective understanding of the universe.

Livio’s work is a testament to the power of mathematical inquiry and the endless pursuit of answers to seemingly impossible questions. Through engaging storytelling and clear explanations, “The Equation That Couldn’t Be Solved” makes a compelling case for the beauty and relevance of mathematics in the modern world.

Here is a pdf version of the book.