1, 2, 3 and to the 4: Getting Back into the Numbers Game

 Posts: 334
 Joined: Mon May 06, 2013 4:43 am
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
One math class that I've constantly used throughout life was algebra. If moving on to advanced classes is the goal, then it will be well worth the effort to go through it, covering all the topics.
quadratic functions, logs and exponents, graphs, system of equations, matrices, inequalities, radical functions,.. etc. I mean every bit of it is very useful.
The rules for the topics covered in algebra will be used over and over again in later advanced classes.
In geometry, I would be sure to understand the anatomy of a circle and the triangle. What's a chord? what's a secant? What makes 454590 and 306090 special? What's a sector? It will also teach you to be able to do simple proofs in a logical manner.
Then I would spend a lot of time learning trigonometry. The anatomy of a periodic function, graphing a periodic function, the ratios in a triangle, the unit circle, the identities, angle formulas and proofs, and the theorems such as de moivre, euler equation.. etc.
The algebra and trig will serve as the foundation for calculus. Now calculus will be different. You will need to start thinking. Definitions will become important. Concepts will become important  i.e. first thing you'll hit, what's a limit? What's continuity? Then you'll really learn what a slope is and that there are different kinds of slopes. They are useful! Things tucked away in alg, trig and geometry will be put to use to form a new concept in the very first chapter of calculus and it goes on from there. Many many ideas in calculus.
All of calculus (usually broken in 3 parts) is useful. Calculus will be what alg, trig and geometry was for later classes.
Then there is diff eq. I strongly suggest DiPrima for that one, it is a great book. Then Linear algebra. These go hand in hand.
Things get really interesting when you get to partial differential equations. This is when you start to realize that the many abstract ideas learned in the past can now be applied to approximate and solve real life problems. The wave equation, the heat equation, just as a small sample to solve things like a drum beat, a heating point source, etc.
And some of these solutions are borrowed to solve problems in finance  black scholes for example and the heat equation.
While these later classes are exciting (for some) the one truly useful class was algebra. It will be used over and over again. Algebra is like the rules for moving the chess pieces and applies to all of the strategies ever concocted.
quadratic functions, logs and exponents, graphs, system of equations, matrices, inequalities, radical functions,.. etc. I mean every bit of it is very useful.
The rules for the topics covered in algebra will be used over and over again in later advanced classes.
In geometry, I would be sure to understand the anatomy of a circle and the triangle. What's a chord? what's a secant? What makes 454590 and 306090 special? What's a sector? It will also teach you to be able to do simple proofs in a logical manner.
Then I would spend a lot of time learning trigonometry. The anatomy of a periodic function, graphing a periodic function, the ratios in a triangle, the unit circle, the identities, angle formulas and proofs, and the theorems such as de moivre, euler equation.. etc.
The algebra and trig will serve as the foundation for calculus. Now calculus will be different. You will need to start thinking. Definitions will become important. Concepts will become important  i.e. first thing you'll hit, what's a limit? What's continuity? Then you'll really learn what a slope is and that there are different kinds of slopes. They are useful! Things tucked away in alg, trig and geometry will be put to use to form a new concept in the very first chapter of calculus and it goes on from there. Many many ideas in calculus.
All of calculus (usually broken in 3 parts) is useful. Calculus will be what alg, trig and geometry was for later classes.
Then there is diff eq. I strongly suggest DiPrima for that one, it is a great book. Then Linear algebra. These go hand in hand.
Things get really interesting when you get to partial differential equations. This is when you start to realize that the many abstract ideas learned in the past can now be applied to approximate and solve real life problems. The wave equation, the heat equation, just as a small sample to solve things like a drum beat, a heating point source, etc.
And some of these solutions are borrowed to solve problems in finance  black scholes for example and the heat equation.
While these later classes are exciting (for some) the one truly useful class was algebra. It will be used over and over again. Algebra is like the rules for moving the chess pieces and applies to all of the strategies ever concocted.
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
A necklace shattered when sleeping together If ⅙ reached the floor, ⅕ lay upon the bed, the girlfriend caught ⅓ of them, ⅒ were salvaged by the boyfriend while six were still stringed. How many pearls did we start out with?

 Posts: 334
 Joined: Mon May 06, 2013 4:43 am
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
If these were the expensive faux pearls then I'd say just take the loss and move on, and chalk it up to an experience of what not to do for posterity.
(Besides, the question could have been simpler: Only 6 pearls were left after 80% of the pearls went missing when a necklace broke. How many pearls were there? 6/20 = x/100 ; x=30 or; .2x=6 x=30 etc. etc...
But probably the easiest is: if 6 is 20% then 100% is 5 times that so there were 30 pearls.)
Yes, I've been cooped up for a while now... (thanks for the humor!)
(Besides, the question could have been simpler: Only 6 pearls were left after 80% of the pearls went missing when a necklace broke. How many pearls were there? 6/20 = x/100 ; x=30 or; .2x=6 x=30 etc. etc...
But probably the easiest is: if 6 is 20% then 100% is 5 times that so there were 30 pearls.)
Yes, I've been cooped up for a while now... (thanks for the humor!)
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
Tbh I only paraphrased his 870 year old poem https://youtu.be/WoJGJeOyLEc

 Posts: 334
 Joined: Mon May 06, 2013 4:43 am
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
I love poems...
Here lies Diophantus,' the wonder behold.
Through art algebraic, the stone tells how old: 'God gave him his boyhood onesixth of his life, One twelfth more as youth while whiskers grew rife;
And then yet oneseventh ere marriage begun; In five years there came a bouncing new son.
Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him.
After consoling his fate by the science of numbers for four years, he ended his life.
A toast to algebra
Here lies Diophantus,' the wonder behold.
Through art algebraic, the stone tells how old: 'God gave him his boyhood onesixth of his life, One twelfth more as youth while whiskers grew rife;
And then yet oneseventh ere marriage begun; In five years there came a bouncing new son.
Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him.
After consoling his fate by the science of numbers for four years, he ended his life.
A toast to algebra
An equation is meaningless unless it expresses a thought of god
Nice!
(1/6 + 1/12 + 1/7)x +5+y+4=x
Putting his son's age y=x/2 above
11x/28 + 9 + x/2=x
(11/28 1 +1/2)x = 9
x=84, y=42
To the seed of mathematics!
Discussing death of mathematicians, today is the 100th death anniversary of Ramanujan although he died at 33.
(1/6 + 1/12 + 1/7)x +5+y+4=x
Putting his son's age y=x/2 above
11x/28 + 9 + x/2=x
(11/28 1 +1/2)x = 9
x=84, y=42
To the seed of mathematics!
Discussing death of mathematicians, today is the 100th death anniversary of Ramanujan although he died at 33.
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
P.S.
Thanks to everyone who contributed suggestions.
I decided to go with khan academy. I started at the most basic level and took subject tests until I got to something where I didn't know literally every answer. This turned out to be Algebra I. I started at the beginning of Algebra, looked at the list of videos and watched videos which seemed interesting. I'm also taking every test along the way and will watch videos/ explore concepts I don't understand (currently I'm still in the beginning of algebra and haven't found a problem that I don't know the answer to yet). This is not the quickest way to do it, but I'm ok with that. It also starts out with a bunch of easy wins.
I'm sure I'll have questions on how to decide what to study when I'm done with "basics" (I think Khan Academy goes through differential equations). Does anyone have any advice on how to make sure I'm learning how to use this stuff and not just learning it in the sterile "school" environment?
Thanks to everyone who contributed suggestions.
I decided to go with khan academy. I started at the most basic level and took subject tests until I got to something where I didn't know literally every answer. This turned out to be Algebra I. I started at the beginning of Algebra, looked at the list of videos and watched videos which seemed interesting. I'm also taking every test along the way and will watch videos/ explore concepts I don't understand (currently I'm still in the beginning of algebra and haven't found a problem that I don't know the answer to yet). This is not the quickest way to do it, but I'm ok with that. It also starts out with a bunch of easy wins.
I'm sure I'll have questions on how to decide what to study when I'm done with "basics" (I think Khan Academy goes through differential equations). Does anyone have any advice on how to make sure I'm learning how to use this stuff and not just learning it in the sterile "school" environment?
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
My advice is to move on to the “test” and start applying what you know to the realworld problems around you. Realizing that not every problem has a solution, or that you might not have the right tools or experience. Make the attempt and learn from it. Ask for help if needed. Choose problems that are consequential or interesting. The extra motivation might be needed to get to the finish line.
If you don’t know where to start, try revisiting this math problem of yours with fresh eyes... answer key provided viewtopic.php?p=197031#p197031
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
Yeah, not sure how much help I can be here. I always found computation the boring part. The TiNe approach is more along the lines of..
1. Parse an example problem.
2. Read about the abstractions being used.
3. Parse a more generalized version of the problem.
4. Read about the abstractions being used.
5. Find additional resources that stem from such abstractions (history, applications, etc.)
6. Think about alternative representations or about the justification for why such abstractions exist.
.
.
.
11. Discover relationships between those abstractions and seemingly unrelated abstractions.
.
.
.
16. Find out that everyone solving real problems just uses a program to hide those abstractions.
17. [if required] Finish assigned HW ASAP. Probably making careless errors along the way.
Proofs were a bit more interesting because they fit better with my learning processes. Mostly, math for me was just a brainstorming enhancer. It might take me an hour now after a few years to derive(*) an integration trick for instance, but all the peripheral thinking I did has been compounded/expanded upon.
(*) By using my intuition to the find muscle memory complexes burred in my mind. So I agree with Jacob that some actual computation is probably worthwhile even for INTP's.
1. Parse an example problem.
2. Read about the abstractions being used.
3. Parse a more generalized version of the problem.
4. Read about the abstractions being used.
5. Find additional resources that stem from such abstractions (history, applications, etc.)
6. Think about alternative representations or about the justification for why such abstractions exist.
.
.
.
11. Discover relationships between those abstractions and seemingly unrelated abstractions.
.
.
.
16. Find out that everyone solving real problems just uses a program to hide those abstractions.
17. [if required] Finish assigned HW ASAP. Probably making careless errors along the way.
Proofs were a bit more interesting because they fit better with my learning processes. Mostly, math for me was just a brainstorming enhancer. It might take me an hour now after a few years to derive(*) an integration trick for instance, but all the peripheral thinking I did has been compounded/expanded upon.
(*) By using my intuition to the find muscle memory complexes burred in my mind. So I agree with Jacob that some actual computation is probably worthwhile even for INTP's.

 Posts: 105
 Joined: Tue Oct 15, 2019 3:13 pm
Re: 1, 2, 3 and to the 4: Getting Back into the Numbers Game
The International Math Olympiad is designed for high school students and has interesting problems to exercise your math skills or just think about. Or you could be like the North Koreans and cheat.
Here's one of them: Six points are chosen on the sides of an equilateral triangle ABC: A1,A2 on BC, B1,B2 on CA and C1,C2 on AB, such that they are the vertices of a convex hexagon A1A2B1B2C1C2 with equal side lengths. Prove that the lines A1B2, B1C2 and C1A2 are concurrent.
Here's one of them: Six points are chosen on the sides of an equilateral triangle ABC: A1,A2 on BC, B1,B2 on CA and C1,C2 on AB, such that they are the vertices of a convex hexagon A1A2B1B2C1C2 with equal side lengths. Prove that the lines A1B2, B1C2 and C1A2 are concurrent.