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I am interested in improving my mathematical abilities and understanding. I wonder what it means to be good at mathematics and what skills are essential for to be good at mathematics in general . of all the skills I think that creating many steps in my head faster and more accurate are very important .

I am interested in learning how to perform arithmetic operations (such as addition, subtraction, multiplication, division, and calculating multiple steps of a problem) faster and more accurately in my head, without using a calculator or paper. I think that this can help me develop my logical thinking and concentration. It can also help me improve my math skills, as I can calculate multiple steps in a shorter time. This will reduce the time that I will spend on a certain approach in a proof or a problem wont lead anywhere and won't work. I think this skill will enhance my intuition in math and make me better at math in general. I have always been fascinated by the ability to solve problems like integrals in your head, without writing anything down. I think this is a very impressive and useful skill.

However, I am not sure how to practice and improve my mental arithmetic skills.I also don’t know what techniques or strategies are effective for different types of problems. For example, how can I quickly multiply two-digit numbers, or divide by fractions, or find the square root of a number ,or more importantly do multiple steps for problems like integrals in my head ?

Therefore, I would like to ask for some book recommendations or sources on how to learn and improve mental arithmetic and mental math skills. What are some books, websites, courses, or other resources that can help me?

pie
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    For finding the square root see for example this post. More links can be found at wikipedia. – Dietrich Burde Sep 19 '23 at 16:15
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    I find this admirable, and I'd like to encourage you. However, let me humbly recommend you practice a skill that isn't so easily served by ubiquitous calculators. In the long run, you'll find it a better use of your time and effort (and will impress more people!). – David G. Stork Sep 20 '23 at 06:10
  • @DavidG.Stork Thank you for your kind words and encouragement. I want to ask you what kind of skills do you recommend me to practice? – pie Sep 20 '23 at 06:47
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    Learn another language, master a non-traditional branch of mathematics (e.g., Transform theory, differential topology, ...), learn public speaking, learn a science of mainly intellectual interest, read great novels, learn to play an instrument... – David G. Stork Sep 20 '23 at 17:39
  • @DavidG.Stork I think this skill is very important and will enhance my intuition in math and make me better at math in general as I can calculate multiple steps in a shorter time. This will reduce the time that I will spend on a certain approach in a proof or a problem wont lead anywhere and won't work – pie Sep 23 '23 at 17:14
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    @pie You may have convinced yourself that these skills will help you in the long run to enhance your mathematical thinking and proofs, but in practice it is rarely the case. In any case though, I don't think there is really a good book for learning how to do mental calculations "better". Instead I think, like all things, it is practice that will take you there. I think books about problem solving, like Polya's, might serve you much better in this regard. – V.S.e.H. Sep 23 '23 at 17:40
  • @V.S.e.H. I appreciate your advice on practising problem solving and reading books like Polya’s. However, I am still confused about what it means to be good at math. What skill should I practice to be good at math? I mean, what should I learn if solving many problems in shorter time is not what makes me a good at math? Then what would make me a good math? what skill should I improve on? I know that people differ in their talent and hard work and I am not very talented person so I want to enhance my skills – pie Sep 25 '23 at 12:12
  • also I don’t know what should I do when I solve problem books like the one you suggested. What should I do? Should I memorise the techniques or try to understand the concepts behind them? How can I improve my mathematical thinking and proofs? Please help me understand better. Thank you. – pie Sep 25 '23 at 12:12
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    I can't suggest a technique that works in general, because something like this doesn't exist imho. You can't expect to be good in math without going through the pain and suffering that other mathematicians go through, as there is no "fast" way. If you want to be good in math, pick a topic, read books/papers about it, and do solve as many exercise problems as you can. Then do the same process again with other topics. As starting topics, I suggest something that helps you build your fundamentals, like real analysis, topology, abstract algebra, etc. For what is worth, that is what works for me. – V.S.e.H. Sep 25 '23 at 13:16
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    How about Arthur Benjamin's books? – Lucky Chouhan Sep 25 '23 at 13:30

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I recommend the book

Vedic Mathematics - Bharati Krishna Tirthaji Maharaja .

It is one of the best books on Mental calculation. The book is primarily a collection of tricks that can be applied in elementary arithmetic and algebra, to make faster mental calculations.

MathRookie2204
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I appreciate your interest in mental mathematics. TBH I have never learned mental mathematics and as being a junior practitioner, I never required deep mental mathematical abilities. And, to address your question, I searched internet to find something better for you and what I find out is a book which I have never read but the title suggests that it might help you. I'd recommend you to go through its content first and then buy a copy if you think it can help you.

"Speed Mathematics Simplified" by Edward Stoddard

Moreover, I found a simple youtube playlist that might help you.

Learn lightning Fast Mental Maths by mathOgenius YT channel
AbdurRehman
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I've found the following book very helpful: https://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401

It covers a few core ideas that will help you rapidly do mental calculations.

However, as mentioned in the comments, blazing mental calculation skills have little benefit for understanding and being good at mathematics. Where I find these skills most useful is rapidly sense-checking calculations or guesstimating quantities (e.g., "back of napkin" calcs).

So mental math will help a lot in data analysis but less so in actual mathematics.

Annika
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  • I didn't understand what are these skills ? can you please till me more about what the skills I need for pure math ?and I also asked about mental math and performing calculation of many steps on my head – pie Sep 29 '23 at 02:08
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    @pie - I gave a reference for mental math above, as you requested in your post. Skills for pure math are harder to define, but involve ability to abstract and recognize different ways of expressing things. I myself am not super strong in pure math, but applied math I'm decent at. For applied math, its a lot about abstracting and being able to understand the size of different effects so you know what needs to be modeled and what can be ignored. – Annika Sep 29 '23 at 03:39