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 · 51-55, 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 Ah! The slang words!

Well, yes we still use "quid" (£1 - now a coin rather than note), "Fiver" for £5, "Tenner" for £10 note.

The £20 note does not have its own nickname. Nor does the £50 note though it's very rare to see one of those. I never have.

The pre-decimal slang or colloquial names were "tanner" (6d, or six penny, coin), "bob" for a shilling.
.

I don't know what our coins are made from these days. I tested one of each except the 50p piece, as I don't have one of those at the moment.

1p and 2p: resemble copper but strongly magnetic, so probably copper-plated mild-steel.

5, 10, 20 p "silver" coins: all non-magnetic. Stainless-steel?

£1 coin, which has a silvery inner disc surrounded by what looks like brass: very weakly magnetic so might be of stainless-steel of a different basic type.

£2 coin: similar in appearance to the £1, though larger: less magnetic still.

That was with an old hard-drive magnet, so quite powerful.

'''

It's an oddity of colloquial speech that although it is plural the word "Pence" is used for just one penny; but that only started with the introduction of decimal currency calling the new £1/100 coin, a "New Penny" - when almost everything already cost at least some pennies / pence.

Britain abandoned the gold standard a long time ago, too. I think our Government does issue bonds but in a controlled way, to avoid inflation. I don't really know how they work.

JoyfulSilence · 51-55, M

@ArishMell

I Googled it earlier, and due to the price of copper, pennies are now been copper coated zinc, like 95% zinc. They said do not swallow one because the zinc is bad in stomach acid. Yikes!

ArishMell · 70-79, M

@JoyfulSilence Really? Not in the UK. Zinc is not magnetic, but the 1p and 2p coins I tested are as magnetic as mild steel. They might be plated with zinc then copper: I don't know if that combination is possible.

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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