Book Review: The Humongous Book of Calculus Problems

I bought this book a year ago as a refresher of Calculus and as of now, I am almost finished reading it. I think what separates this book from the rest are the numerous worked examples (well, 1000 of them) with detailed solutions and explanations. Additional pointers and explanations in layman’s words are provided as notes.

This book has 565 pages containing 28 chapters. The first 8 chapters contain a review about equations, polynomials, functions, and trigonometry, while the remaining chapters discussed topics in Calculus I and II: Limits, Differentiation, Integration, Parametric and Polar Equations, Sequences and Series. As a bonus, a chapter on Differential Equations is also included.  Continue reading

How to Change Number Bases Part 1

I have already discussed clock arithmetic,  modulo division, and number bases. We further our discussion in this post by learning how to change numbers from one base to another.

The number system that we are using everyday is called the decimal number system or the base 10 number system (deci means 10). It is believed that this system was developed because we have 10 fingers.

In the base 10 system, the digits are composed of  0 up to 9. Adding 1 to 9, the largest digit in this system, will give us 10. That is, we replace 9 in the ones place with 0, and add 1 to the tens place which is the next larger place value.

Another way to write a number in base 10 is by multiplying its digits by powers of 10 and adding them. For example, the number 2578 can be rewritten in expanded form as

2(10^3) + 5(10^2) + 7(10^1) + 8(10^0)Continue reading

Banach-Tarski Paradox and A Great Discussion About Infinity

This video is a rich discussion about about infinity. It starts from the famous infinite chocolate problem, countable infinity, uncountable infinity, Cantor’s diagonalization system, Hilbert’s hotel, and the Banach-Tarski Paradox. Although the last part is a bit too much for the level of this blog, the explanations before it of the concepts before it are extremely clear and helpful.

I had several discussions about infinity in this blog particularly the content of Counting the Real Numbers, The Grand Hotel Paradox, and One to One Correspondence. You might want to check them out.

Understanding If-Then Statements

If I get my bonus, then I will buy a car.
If I become 30, then I will get married.
If x \geq 3, then x^2 \geq 9.

The statements above are examples of an if-then statement. An if-then statement is also called a conditional statement. A conditional statement is of the form if p then q where p and q are statements and in which q follows from p. As shown above, conditional statements can be seen in real life or in mathematics.

In a conditional statement, the statement following if is called the antecedent (or hypothesis), while the statement following then is called the consequent (or conclusion).

Conditional statements have their truth values just like, and, or and not. To examine their truth values, let us consider the third conditional statement above. In this statement, we can assign p as x \geq 3  and q as x^2 \geq 9Continue reading

Hundreds of Combinatorics Video Lectures

If you are taking mathematics courses related to combinatorics or probability, I found a page from UCLA containing hundreds of video lectures on the said subjects. It is a collection of lectures from 1993 up to the present and includes lectures from famous mathematicians such as Terence Tao.

For more free video lectures, visit the All for Free page of Math and Multimedia. It also contains posts which links to hundreds of free ebooks and software.

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