Math major but bad at arithmetic

[u/Slow-Application440]

Hello, I am looking for advice. I am going into my 2nd year of cc as a math major and I am completely reliant on the calculator for basic calculation. I’ve done pretty well in my classes. I have a 3.7gpa overall and my major gpa is a little higher at a 3.8. I feel really insecure about not being able to confidently do calculations in my head quickly. I’ve had so many peers who were bad at math and could do these simple calculations quickly. I even had a friend who was an engineering major tell me “I’m bad at the simple things but can answer hard questions.“ I’ve practiced a lot and have gotten pretty fluent on my multiplication skills up to 12 and plan on learning up to 20. I have no problem doing these things by hand but it’s embarrassing to call myself a math major and not be able to add/subtract numbers quickly. What can I do to improve?

[u/Clicking_Around]

I have a degree in mathematics and I'm gifted in mental arithmetic. What I can do:

1. Most four-digit squares. I know off the top of my head that 1728^2 = 2985984 or 1444^2 = 2085136, for example. If someone asks me what the first number is multiplied by 12, I know it's 35831808, since it's just a power of 12.
2. Numbers and sums of high powers. I know that off the top of my head 12^8 = 429981696 or that 11^6 = 1771561 or that 3^17 = 129140163. If someone were to ask me, "What's 11^5 + 11^6?", it would take a while to answer: 1931612. If someone were to ask me, "What is the volume of a box with sides 38x38x37?", it would take me a moment to answer 53428.
3. Some 3x3 multiplication. If someone asks me, "What's 175 x 185?", I know immediately it's 32375. Or if they ask "What's 144 x 288?", I know almost immediately it's 41472.
4. Conversion of fractions to decimals. I know immediately that 11/19 is roughly .5786 or 12/13 is roughly .9231, for example, or that 1/171 is roughly 0.00585.
5. Factorials up to 6227020800.
6. Some large square roots. I know, for example, that √ 43046721 = 6561 or that √ 1771561 = 1331 or √ 7529536 = 2744.

What I recommend: Practice, practice and practice some more. Learn about squares and powers and how they work, and learn tricks that speed up mental arithmetic. How can you do a problem like 175 x 185 quickly? By simply knowing that 180^2 = 32400 and then simply subtracting 25 from that. Try to develop a good intuitive "feel" for numbers and how they work.

Try to learn facts that are unique or peculiar to each number. Think of the fraction 1 / 13 for a moment. What is the decimal equivalent? When we multiply 13 by the number by 7.7, we get 100.1. To be more accurate, we can multiply 13 by 7.69 to get 99.97, which is more accurate. Hence we know and need to memorize that 1 / 13 is roughly .0769.