Discovering the wonderful world of logarithms has made me a happy camper. Now I can do calculations that would have required a calculator before.
I printed out a direct lookup table with no x-values skipped that ranges from 1 to 10,000 inclusive, as well as a mantissa-based table that goes up to three decimal places, ranging from 0.101 to 9.999 (I have a much more precise mantissa table with x-values that go to six decimal places, but it’s unwieldy due to its size, so I have to use it on my phone.)
The direct lookup table is good when I want to quickly get the logarithm of a number, while the mantissa-based table is great for looking up answers after I’ve figured them.
Welcome to logarithms! Before calculators (1975) we had slide rulers. Understanding logarithms was so easy and natural on a slide rule. At one time I loved thinking in logarithms and solving problems in my head! (back then, we only needed 3 significant digits)
So many terrible teachers out there. Often just one bad teacher is enough to derail someone, since it’s not like English, history, or other subjects where gaps don’t matter so much. Mathematical gaps matter. I hope to publish a book to help someday.
I don’t know if it is matter of reading on my phone, but a lot of “+” were dropped in the equations, which made some of the proofs initially hard to follow. Did anyone else see that? I saw a^2-b^2=(ab)(a-b), and I was left scratching my head for a second. Everything handwritten is perfect. I’m going to work on the Putnam problem before reading Holly’s proof.
I did multiple test emails and had several smart-but-not-math-loving people read this over, and nobody else mentioned this, so I'm going to assume it was your phone unless anyone else comments that they had the same issue. Thanks for mentioning it!
Jesus Fucking Christ, man. Are you trying to give me a panic attack? What, SPECIFICALLY, are you talking about? The only image files are my handwriting. How did I make typos in my own handwriting? Much less "big" ones?!?
I didn’t mean to annoy you; the part with a^2 - b^2 = (ab)(a - b). That’s what I see there; it’s where you’re explaining how the squares of all primes greater than 3 leave a remainder of 1 when divided by 24 (an awesome math fact, by the way.)
I still need to know what kind of phone you are using the app on. I don't beg for specifics because I enjoy begging. It's because that's the only hope of getting them to fix it.
What specific device and interface (app? Safari? Chrome? Something else?) does it look wrong on? I need to report to Substack but I cannot do that without *specific* information. Thanks!
Hi Holly. I really tried to follow this but had to give up. I’m pretty good at mental maths but what I realise is that I don’t understand the specialist maths notation. When I learned maths (many years ago) x was the sign for multiply. It now seems to be a dot? Once we started on the ab’s and brackets I was lost. Maybe you could do a special on basic maths notation for non maths people?
When x is used as a variable or part of a problem, as in this one, using it as the sign for multiplication creates issues, so using a dot promotes clarity. I will give some thought to that, thanks.
I apologize in advance for my ignorance, and you may have covered it in a section of "How Not To...", which I haven't gotten to yet, but modular arithmetic, with which I am unfamiliar, looks somewhat like using a base other than 10, a practice I was introduced to so long ago that 'the face rings a bell, but I can't recall the name.' A simple, "you have the wrong end of the stick, and you are holding the stick wrong" is sufficient if I am way off base, as it were. But I am hoping that your non-mathish readers, including me, would get some benefit if it prompts a post from you, eventually. Thanks.
Your mind is working the right way for that to be what it makes you think of, even though it's not the same thing. I will write a post on it eventually, yes!
Hi, many thanks for the wonderful post, it made my day! An alternative proof for the factor 8, possibly easier to understand for non-math people, is analogous to your proof for the factor 3.
p-1, p, p+1, p+2 are 4 consecutive numbers, so one of them is divisible by 4, and one other is divisible by 2. Since both p and p+2 are odd, that means that p-1 and p+1 together have a factor of 8.
Yes. I very nearly used this, but I thought it didn’t illuminate the thought process for a number theory proof as well. One of my goals in life is to get as many people as possible over their mathphobia, partly because they’ll be happier but mostly because then journalists will have a harder time lying to us.
Funny note: Some years ago I was playing with Python on my laptop and wrote a little program to calculate prime numbers. I started it and let it run for a while. It ran so fast that it burned up my CPU and wrecked the laptop! I have to admit I'm a little proud of the fact that I wrecked a laptop with a little Python code.
Discovering the wonderful world of logarithms has made me a happy camper. Now I can do calculations that would have required a calculator before.
I printed out a direct lookup table with no x-values skipped that ranges from 1 to 10,000 inclusive, as well as a mantissa-based table that goes up to three decimal places, ranging from 0.101 to 9.999 (I have a much more precise mantissa table with x-values that go to six decimal places, but it’s unwieldy due to its size, so I have to use it on my phone.)
The direct lookup table is good when I want to quickly get the logarithm of a number, while the mantissa-based table is great for looking up answers after I’ve figured them.
Welcome to logarithms! Before calculators (1975) we had slide rulers. Understanding logarithms was so easy and natural on a slide rule. At one time I loved thinking in logarithms and solving problems in my head! (back then, we only needed 3 significant digits)
Yup, using logarithms is a lot of fun. If only I could print a five-digit table, though.
Really fascinating for a numerate non-mathematician. The question is how to engage math-phobes before they switch off.
So many terrible teachers out there. Often just one bad teacher is enough to derail someone, since it’s not like English, history, or other subjects where gaps don’t matter so much. Mathematical gaps matter. I hope to publish a book to help someday.
I don’t know if it is matter of reading on my phone, but a lot of “+” were dropped in the equations, which made some of the proofs initially hard to follow. Did anyone else see that? I saw a^2-b^2=(ab)(a-b), and I was left scratching my head for a second. Everything handwritten is perfect. I’m going to work on the Putnam problem before reading Holly’s proof.
I did multiple test emails and had several smart-but-not-math-loving people read this over, and nobody else mentioned this, so I'm going to assume it was your phone unless anyone else comments that they had the same issue. Thanks for mentioning it!
It’s in the image files showing the difference of squares. You may have made big typos there.
Jesus Fucking Christ, man. Are you trying to give me a panic attack? What, SPECIFICALLY, are you talking about? The only image files are my handwriting. How did I make typos in my own handwriting? Much less "big" ones?!?
I didn’t mean to annoy you; the part with a^2 - b^2 = (ab)(a - b). That’s what I see there; it’s where you’re explaining how the squares of all primes greater than 3 leave a remainder of 1 when divided by 24 (an awesome math fact, by the way.)
(This note is for anyone who comes across this thread later.)
Everything has been resolved. It was a simple misunderstanding due to LaTeX not functioning correctly on phones when using the Substack app.
I still need to know what kind of phone you are using the app on. I don't beg for specifics because I enjoy begging. It's because that's the only hope of getting them to fix it.
I was looking at it on an iPhone 13 Pro.
What specific device and interface (app? Safari? Chrome? Something else?) does it look wrong on? I need to report to Substack but I cannot do that without *specific* information. Thanks!
For me, it is iOS 17.5.1, Substack app.
That is exactly what I see. The plus sign + is missing in the factoring equations. Left me completely stumped until I realized what wasn’t there.
I’m using the Substack app on iPadOS 17.6
Hi Holly. I really tried to follow this but had to give up. I’m pretty good at mental maths but what I realise is that I don’t understand the specialist maths notation. When I learned maths (many years ago) x was the sign for multiply. It now seems to be a dot? Once we started on the ab’s and brackets I was lost. Maybe you could do a special on basic maths notation for non maths people?
When x is used as a variable or part of a problem, as in this one, using it as the sign for multiplication creates issues, so using a dot promotes clarity. I will give some thought to that, thanks.
Wow! Thanks. Truly interesting.
I apologize in advance for my ignorance, and you may have covered it in a section of "How Not To...", which I haven't gotten to yet, but modular arithmetic, with which I am unfamiliar, looks somewhat like using a base other than 10, a practice I was introduced to so long ago that 'the face rings a bell, but I can't recall the name.' A simple, "you have the wrong end of the stick, and you are holding the stick wrong" is sufficient if I am way off base, as it were. But I am hoping that your non-mathish readers, including me, would get some benefit if it prompts a post from you, eventually. Thanks.
Your mind is working the right way for that to be what it makes you think of, even though it's not the same thing. I will write a post on it eventually, yes!
Disclaimer:
I’m not a math person so I apologize in advance if this joke is inaccurate and/or not funny.
PSA: Eating too much cake is the sin of gluttony. However, eating too much pie is okay because the sin of pi is always zero.
A classic! I will do a post explaining this joke someday.
Thank you, I need all the help I can get. 😜
Hi, many thanks for the wonderful post, it made my day! An alternative proof for the factor 8, possibly easier to understand for non-math people, is analogous to your proof for the factor 3.
p-1, p, p+1, p+2 are 4 consecutive numbers, so one of them is divisible by 4, and one other is divisible by 2. Since both p and p+2 are odd, that means that p-1 and p+1 together have a factor of 8.
Yes. I very nearly used this, but I thought it didn’t illuminate the thought process for a number theory proof as well. One of my goals in life is to get as many people as possible over their mathphobia, partly because they’ll be happier but mostly because then journalists will have a harder time lying to us.
Love this post! Very understandable explanations!
Funny note: Some years ago I was playing with Python on my laptop and wrote a little program to calculate prime numbers. I started it and let it run for a while. It ran so fast that it burned up my CPU and wrecked the laptop! I have to admit I'm a little proud of the fact that I wrecked a laptop with a little Python code.