Did The Light Dawn Decades Later?

Mathematics was my weakest and least enjoyable subject at school, due my own inability to learn it, by it seeming merely a school-leaving examination subject, and quite frankly by two bad teachers.

In the many years after leaving school, I came to use some topics, and even managed to understand some, in unexpected ways through work and hobbies.

Particularly....

- Trigonometry and pi in various applications, but I had no difficulties with basic geometry and mensuration;

- Differentiation by a sheer fluke. Attending a geology-club lecture on rivers, something made me write the simple formula in calculus notion and I suddenly twigged this was simple differentiation, hence what differentiation really does! Integration remains a mystery although I know what it does graphically.

- Logarithms, also indirectly. I realised I had to understand logarithms to understand decibels, when I entered work that used them. I am old enough for logarithms to have been a general-purpose arithmetical tool in my school years, on the verge of electronic calculators appearing, but though I could use log. tables for that purpose I did not understand them. I can still use them, and slide-rules, for arithmetic.

- Algebra. Improving! If you are weak at algebra obviously you will be at Maths generally, because you can no more have mathematics without algebra than you can have arithmetic without numbers. With a few exceptions of course, such as Pure (or Euclidean) Geometry which does not use formulae, values and calculations... but needs a good memory to remember many Theorems!.

Anyone else had similar epiphanies?

+-=*

I should explain that the UK's school system teaches Mathematics as a single cohesive curriculum subject of many topics including those above. It does not break Maths into separate curriculum subjects or courses for each topic, as American schools appear to do. (Although no-one has confirmed that deduction, neither has anyone said I am mistaken!)

The school-leaving exam was the General Certificate of Education "Ordinary Level".

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JoyfulSilence · 46-50, M

Integration is a way of adding up N small things, as their size goes to zero and N goes to infinity.

Int( f(x) , x=a..b )

= lim(n-->inf) of
(b-a)/n *
Sum( f( x(k) ) , k=1..n )
where
x(k) = a + (b-a)* k/n

That is, divide the interval [a,b] into n intervals, and compute that sum. Then take the limit of the sum as n goes to infinity.

Just like a derivative is the slope between nearby points on a curve, as the points get closer and closer together.

df/dx = f'(x) = lim(h-->0) of
(f(x+h)-f(x))/h

They are related. Suppose we are given a function F(x) such that
F'(x) =f(x).
Then
F(b)-F(a)
= Int( f(x) , x=a..b )

Also, suppose we are not given F, but we define it as:
F(x) = Int( f(t) , x=a..t)
for some arbitrary a.
Then it follows that
F'(x) =f(x), for all x and a.

The natural log is just this integral:
ln(x)= Int( 1/t , t=1..x ), fir x>0.

It can be shown to have log properties:
ln(x*y)=ln(x)+ln(y)
p*ln(x)=ln(x^p) for all rational p.

It is an increasing, and hence invertible function. Then there exist a number, call it "e", such that ln(e)=1.

Let "exp" be the inverse function.
That is,
exp(ln(x))=x, fir x>0, and
ln(exp(x))=x, for all x.

exp has exponential properties. We know:
ln(exp(x)*exp(y))
=ln(exp(x))+ln(exp(y))
=x+y
Hence:
exp(x+y)= exp(x)*exp(y)

Also, for all rationals p
ln((exp(x))^p)
=p*ln(exp(x))
=p*x
Hence:
(exp(x))^p=exp(x*p)

For irrational r, we define
x^r=exp(r*ln(x))
Suppose we have a sequence p(n) of rationals so that p(n)-->r
Then
x^r=exp(r*ln(x))
=exp(lim(p(n))*ln(x))
=lim exp(p(n)*ln(x))
=lim exp(ln(x^p(n)))
=lim x^p(n)
etc.

Since, ln(x)= Int( 1/t , t=1..x )

Then ln'(x) = 1/x.

Let f(x)=exp(x). Then
ln(f(x))=x.

Differentiating both sides, and using the chain rule: gives:

ln'(f(x)) f'(x) = 1.
(1/f(x))*f'(x)=1
exp'(x)=f'(x)=f(x)=exp(x)

Neat!

Also, the geometric mean of a set of numbers is the same as exp(A) where A is the arithmetic mean of their natural logs.

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ArishMell · 70-79, M

@JoyfulSilence I fear Public Inquiries sometimes end up the same way: lots of recommendations, followed by lots or prattling about "leaning lessons and putting procedures in place to ensure it can never happen again".

Sometimes that does happen but I wonder how many "lessons" become forgotten in a pile of later work or just inertia.

We started this exchange about mathematics.

I have encountered something that seems to defy any known arithmetical analysis yet appears to be a special type of circular slide-rule for some long-ago trade or measurements-conversion; and take some comfort for people with more numerate brains than mine being unable to work out any pattern to it!

Now, my work sometimes involved Excel so I became moderately fluent with that, at a fairly simple arithmetical level, and now wonder if shovelling this "slide-rule" into a spreadsheet and testing the scales with various sums might reveal anything.

I can still use a conventional slide-rule, with a bit of revision, but this one, made in 1908, defies all logic!

JoyfulSilence · 46-50, M

@ArishMell

I saw that other post, with the round thingy. I have no clue what it is, either

But I have never even held a slide rule. I would have no clue how to use it. We all had hand held electronic calculators when I was young. I still have my Texas Instruments solar powered scientific calculator from high school. Still works!

But for my job I use Excel, or else write a program (usually in a language/application) called SAS, which is itself a dinosaur.

ArishMell · 70-79, M

@JoyfulSilence Ah well, I am a generation older than you!

Electronic calculators were just beginning to appear at about the time I left school in 1970, so we were all accustomed to using slide-rules and log. tables for awkward calculations in Maths and Physics homework.

I still have two slide-rules. One was my own anyway, and I inherited the other from my Dad, who had been a Chartered Electrical Engineer so much of his work was highly mathematical. (I don't know what he actually did in detail because that was all confidential, but he did have a Degree and I have seen some of his course notes and text-books.)

That type of slide-rule is not longer made as far as I know, but special ones for particular trades are, and many of those are circular rather than linear like a general-purpose arithmetical slide-rule.

It amused me a bit when I see things described as "dinosaurs". The real dinosaurs as an entire group were very successful animals indeed, and their eventual demise was by forces far beyond their control or ability to handle!

If Excel or the SAS language still work as you intend for your purposes, they are hardly "dinosaurs" just because they were developed a few decades ago.

DrWatson · 70-79, M

I had a similar experience with learning Spanish. I was "ok" at it in school, but many years later, as an adult, I decided to try to review it. I got to the point where I picked up a novel which we had read in Spanish class, but this time around I was able to understand and appreciate it a lot better than I ever did.

Then we decided to take a trip to Europe, and I immersed myself in an audio course, and when we went to Spain I was able to hold my own in simple conversations. I never would have been able to comprehend what people were saying to me on the basis of my high school Spanish alone.

Through this all, I was very conscious of the fact that I was not re-learning "from scratch", but that I was building on the foundation I received in school.

ArishMell · 70-79, M

@DrWatson An excellent outcome!

DrWatson · 70-79, M

@ArishMell And likewise with you!

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