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<h1>Math I Ought to Know</h1>
<div class="keepws">
(<span class="bold">notes</span>
(date 2025-06-12)
(category <a href="../../notes.html">science</a>)
(status <span class="status-evergreen">evergreen</span>)
(confidence certain)
(importance 9)
(tags methodology personal learning))
</div>
<p class="preview">a list of textbooks, workbooks, and some notes on the math you ought to know</p>
<div class="prose">
<p>I got this idea from <a href="http://bactra.org/notebooks/math.html">Cosma</a>. There seems to be a very wide range of people who have a fundamental misunderstanding of
what it is that a mathematician does, and how that does or does not relate to the mathematics they desire to Learn. I always reccomend a goal oriented learning.
As an example for something like Competitive Programming, I would reccomend learning Discrete Mathematics, and Algorithms. For something like Data Science,
I would reccomend learning Statistics, and Linear Algebra. Your required prerequisites, and the overall depth of knowledge you will need to achieve vary greatly
depending on your goals. Once you have however identified your goals. The question becomes, what is it that I ought to know? That is what this page is for. A list of
some of the textbooks I believe carry value in each of the categories you could possibly want to Learn.</p>
<p>If you have anything to add please fill out the <a href="/contact">contact form</a>. Be specific, and mention it is for a book suggestion for this exact list.
If you have a good suggestion, chances are it will end up on the list. I don't have enmity towards any book or author. I completely understand that
different people have different learning styles and preferences, and what works for one person may not work for another. That being said, I do believe
that there are certain books that are universally valuable for anyone looking to deepen their understanding of mathematics.</p>
<h2>On The List</h2>
<p>Please take not that this is not a curriculum meant to be followed in any particular order. Rather it is a list of books grouped by category, that I believe
are the best in their respective categories. I have tried to include a wide range of topics, from the most basic to those needng the most prerequisites.</p>
<h2>Mathematical History</h2>
<ul>
<li><a href="/link-or-placeholder">Men of Mathematics</a> by E. T. Bell (1937)</li>
<li><a href="/link-or-placeholder">The World of Mathematics</a> edited by James R. Newman (1956; rev. ed. 1959)</li>
</ul>
<h2>Mathematics Puzzles</h2>
<ul>
<li><a href="/link-or-placeholder">The Moscow Puzzles</a> by Boris A. Kordemsky (1962)</li>
<li><a href="/link-or-placeholder">Mathematical Puzzles: A Connoisseur’s Collection</a> by Peter Winkler (2004)</li>
</ul>
<h2>Arithmetic, Prealgebra, and Algebra</h2>
<ul>
<li><a href="/link-or-placeholder">Basic Mathematics</a> by Serge Lang (1971; 2nd ed. 1975)</li>
<li><a href="/link-or-placeholder">The Art of Problem Solving, Vol. 1: The Basics</a> by Richard Rusczyk & Sandor Lehoczky (2006)</li>
</ul>
<h2>Geometry</h2>
<ul>
<li><a href="/link-or-placeholder">Euclid’s Elements</a> by Euclid (c. 300 BCE; Heath trans. 1956)</li>
<li><a href="/link-or-placeholder">Kiselev’s Geometry: Book I. Planimetry</a> by A. P. Kiselev (1951; Dover repr. 1969)</li>
</ul>
<h2>Trigonometry & Precalculus</h2>
<ul>
<li><a href="/link-or-placeholder">Precalculus: Mathematics for Calculus</a> by James Stewart, Lothar Redlin & Saleem Watson (7th ed., 2015)</li>
<li><a href="/link-or-placeholder">Algebra and Trigonometry</a> by Robert F. Blitzer (5th ed., 2018)</li>
</ul>
<h2>Differential & Integral Calculus</h2>
<ul>
<li><a href="/link-or-placeholder">Calculus, Vol. I: One-Variable Calculus with an Introduction to Linear Algebra</a> by Tom M. Apostol (2nd ed., 1969)</li>
<li><a href="/link-or-placeholder">Calculus</a> by Michael Spivak (4th ed., 2008)</li>
</ul>
<h2>Linear Algebra</h2>
<ul>
<li><a href="/link-or-placeholder">Introduction to Linear Algebra</a> by Gilbert Strang (4th ed., 2009)</li>
<li><a href="/link-or-placeholder">Linear Algebra Done Right</a> by Sheldon Axler (3rd ed., 2015)</li>
</ul>
<h2>Discrete Mathematics</h2>
<ul>
<li><a href="/link-or-placeholder">Discrete Mathematics and Its Applications</a> by Kenneth H. Rosen (7th ed., 2011)</li>
<li><a href="/link-or-placeholder">Concrete Mathematics</a> by Ronald L. Graham, Donald E. Knuth & Oren Patashnik (2nd ed., 1994)</li>
</ul>
<h2>Probability & Statistics</h2>
<ul>
<li><a href="/link-or-placeholder">A First Course in Probability</a> by Sheldon M. Ross (8th ed., 2014)</li>
<li><a href="/link-or-placeholder">Introduction to Probability</a> by Dimitri P. Bertsekas & John N. Tsitsiklis (2nd ed., 2008)</li>
</ul>
<h2>Number Theory</h2>
<ul>
<li><a href="/link-or-placeholder">An Introduction to the Theory of Numbers</a> by G. H. Hardy & E. M. Wright (5th ed., 1979)</li>
<li><a href="/link-or-placeholder">Elementary Number Theory</a> by David M. Burton (7th ed., 2010)</li>
</ul>
<h2>Mathematical Logic</h2>
<ul>
<li><a href="/link-or-placeholder">How to Prove It</a> by Daniel J. Velleman (2nd ed., 2006)</li>
<li><a href="/link-or-placeholder">A Mathematical Introduction to Logic</a> by Herbert B. Enderton (2nd ed., 2001)</li>
</ul>
<h2>Bridge to Higher Mathematics</h2>
<ul>
<li><a href="/link-or-placeholder">Book of Proof</a> by Richard Hammack (2nd ed., 2013)</li>
<li><a href="/link-or-placeholder">Transition to Advanced Mathematics</a> by Douglas Smith, Maurice Eggen & Richard St. Andre (7th ed., 2014)</li>
</ul>
<h2>Abstract Algebra</h2>
<ul>
<li><a href="/link-or-placeholder">Abstract Algebra</a> by David S. Dummit & Richard M. Foote (3rd ed., 2003)</li>
<li><a href="/link-or-placeholder">A First Course in Abstract Algebra</a> by John B. Fraleigh (7th ed., 2002)</li>
</ul>
<h2>Real Analysis</h2>
<ul>
<li><a href="/link-or-placeholder">Principles of Mathematical Analysis</a> by Walter Rudin (3rd ed., 1976)</li>
<li><a href="/link-or-placeholder">Introduction to Real Analysis</a> by Robert G. Bartle & Donald R. Sherbert (4th ed., 2011)</li>
</ul>
<h2>Complex Analysis</h2>
<ul>
<li><a href="/link-or-placeholder">Complex Analysis</a> by Lars V. Ahlfors (3rd ed., 1978)</li>
<li><a href="/link-or-placeholder">Complex Variables and Applications</a> by James W. Brown & Ruel V. Churchill (9th ed., 2013)</li>
</ul>
<h2>Topology</h2>
<ul>
<li><a href="/link-or-placeholder">Topology</a> by James R. Munkres (2nd ed., 2000)</li>
<li><a href="/link-or-placeholder">Introduction to Topology: Pure and Applied</a> by Colin Adams & Robert Franzosa (2nd ed., 2007)</li>
</ul>
<h2>Differential Equations</h2>
<ul>
<li><a href="/link-or-placeholder">Elementary Differential Equations and Boundary Value Problems</a> by William E. Boyce & Richard C. DiPrima (10th ed., 2012)</li>
<li><a href="/link-or-placeholder">Ordinary Differential Equations</a> by Vladimir I. Arnold (3rd ed., 1992)</li>
</ul>
<h2>Numerical Analysis</h2>
<ul>
<li><a href="/link-or-placeholder">Numerical Analysis</a> by Richard L. Burden & J. Douglas Faires (9th ed., 2011)</li>
<li><a href="/link-or-placeholder">An Introduction to Numerical Analysis</a> by Kendall E. Atkinson (2nd ed., 1989)</li>
</ul>
<h2>Functional Analysis</h2>
<ul>
<li><a href="/link-or-placeholder">Introductory Functional Analysis with Applications</a> by Erwin Kreyszig (10th ed., 1978)</li>
<li><a href="/link-or-placeholder">Functional Analysis</a> by Walter Rudin (2nd ed., 1991)</li>
</ul>
<h2>Mathematical Writing</h2>
<ul>
<li><a href="/link-or-placeholder">Mathematical Writing</a> by Donald E. Knuth, Tracy Larrabee & Paul M. Roberts (1989)</li>
<li><a href="/link-or-placeholder">Handbook of Writing for the Mathematical Sciences</a> by Nicholas J. Higham (2nd ed., 1998)</li>
</ul>
<p>Suggestions from tons of mathoverflow, mathstackexhange, mathematics discord, and reddit users have been a massive help.
I will try to link of the relevant threads, and discussion here as I find them.</p>
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