The Manga Guide to Calculus starts off with a brief introduction and review of functions. It then jumps right into derivatives, what they mean, and how to compute them. Integration is then considered along Taylor series and partial differentiation. Throughout the story, the guide frequently references real world applications in economics, physics, and chemistry, and explains problems in these fields through the use of calculus. The author also includes probability, statistics, and trigonometry sections with calculus explanations. Exercises exist at the end of each section for you to complete and solutions are presented at end of the book.

There are a few downsides to the manga style of presentation. Formal proofs and definitions do not lend themselves well to be included, and the book is certainly lacking in this area. There may also be some areas which the reader will have to go over a few times to fully understand and see how the book goes from one idea to the next because of the amount of information being presented.

I would recommend this book to those who want a brief review of calculus, beginners who want context as to its uses, and to those who enjoy reading a good math book. This book is not for those who need a thorough review since many important topics are skipped such as limits, related rates, and volumes of rotations. Overall, Hiroyuki Kojima and Shin Togami did an excellent job in writing and illustrating the book respectively, which makes The Manga Guide to Calculus a very different and attractive learning tool.

In graph theory, it is not very difficult to prove that any planar graph is five colorable. One proof of the theorem relies on the fact that if a graph is a K­₅ it is non-planar. Thus, at least one vertex must not be adjacent to a second and you can alternate between the colors of those two vertices to do the rest of the graph.

In Google’s main logo as seen on google.com, the logo uses blue, red, yellow, and green. Four colors. So far, so good. When you look at the favicon, they use blue, red, yellow, green, and white. Five colors. Some may argue that the white shouldn’t be counted because it is the background color, but I believe that when a logo needs the background as much as it does in Google’s favicon (without it there wouldn’t be a ‘g’) it should be counted.

The logo could easily be adjusted as to only use four colors. For example, the blue parts could be colored red and the white could be changed to blue. Or following along with the use of two blue parts and two red parts in Google’s main logo, the green color be yellow, the yellow in the g could be red and the g could be colored green instead of white. Problem solved.

Even the main logo could also be changed to accommodate for the white background either by changing the color of the ‘o’ or ‘l’ to blue or red and leaving the background white or the background could be changed to the original color of the ‘o’ or ‘l’.

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The problem goes like this: How many fractions are there such that the numerator and the denominator make up the decimal equivalent. Take 5/2 = 2.5 for example. Each number in the fraction is in the decimal equivalent and vice versa.

I’ve been working on this problem off and on for the past couple of days and I don’t think I’m really getting anywhere. My approach has been, a/b = a.b  if b > a and a/b = b.a if a > b. If they are equal, I would think that 1/1 = 1 would be the only solution. Initially, I tried doing a = a.b x b and writing that as (ab + floor(b^2/10)).(b^2 mod 10) but I realized that this method would only really work if b < 10.

So I’m kind of stuck on this problem, but I’m still working. Hopefully I’ll get it soon.