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ArishMell · 70-79, M
'Tis a curious thing... regarding Number Two.
I was introduced to logarithms as arithmetical calculating tools way back in the 4th of Primary School in 1962, when I was still 10; though possibly as our teacher's own extension to the syllabus.
They remained the universal hard-sums technique, along with their slide-rule equivalent, until about the end of the 1960s / early-1970s, when electronic calculators started to appear. One or two fellow-students had them in our last Year or two at school; while most of us were still using slide-rules.
I can still use both log tables and the slide-rule, with a little revision; yet - and this is what is odd - I did not [i]understand[/i] them until well within the last twenty years! And then only when work and a some private study outside that, meant I needed understand deciBels; hence coming back round to understanding what logarithms actually are.
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What does Napier's equation tell us? That to multiply two numbers, add their logarithms; then the product is the antilogarithm of that sum. Divide by subtracting the logs. This extends to it being simple to calculate awkward powers and roots: multiply or divide the log of the original number by the index. (This is ordinary numbers and logs, both to base-10: I don't know my way around Hyperbolic Logarithms and have never used them.)
I do not know if they are still published at all, but at one time you could very easily buy books of look-up tables of logarithms & antilogs, trigonometrical constants, etc. Scientific calculators are programmed with them (or with their generating functions?).
I was introduced to logarithms as arithmetical calculating tools way back in the 4th of Primary School in 1962, when I was still 10; though possibly as our teacher's own extension to the syllabus.
They remained the universal hard-sums technique, along with their slide-rule equivalent, until about the end of the 1960s / early-1970s, when electronic calculators started to appear. One or two fellow-students had them in our last Year or two at school; while most of us were still using slide-rules.
I can still use both log tables and the slide-rule, with a little revision; yet - and this is what is odd - I did not [i]understand[/i] them until well within the last twenty years! And then only when work and a some private study outside that, meant I needed understand deciBels; hence coming back round to understanding what logarithms actually are.
+++
What does Napier's equation tell us? That to multiply two numbers, add their logarithms; then the product is the antilogarithm of that sum. Divide by subtracting the logs. This extends to it being simple to calculate awkward powers and roots: multiply or divide the log of the original number by the index. (This is ordinary numbers and logs, both to base-10: I don't know my way around Hyperbolic Logarithms and have never used them.)
I do not know if they are still published at all, but at one time you could very easily buy books of look-up tables of logarithms & antilogs, trigonometrical constants, etc. Scientific calculators are programmed with them (or with their generating functions?).
ArishMell · 70-79, M
@Sazzio LOL!
Far from it. I was always a slow learner and Maths was one of my weakest subjects - as my post hints.
I understood logarithms eventually, some 30-40 years after leaving school, by a rather indirect route that gave a physical anchor to my thinking.
I had had to think of my own way around the problem - and if there is one thing most mathematics text-books dismally fail to do, it is to [i]explain[/i] the mathematics they illustrate.
Similarly with Calculus.
Although Calculus was taught in the Fifth Form of senior school, as part of the GCE O-Level Mthematics syllabus, I could not grasp Differentiation until a chance encounter in a geology club lecture probably thirty years later. Shown a numerical technique for analysing river gradients, something inspired me to write the simple formula in calculus notation. It dawned on me this was a sort of very basic [i]differentiation,[/i] and so finally twigged what that term actually means, hence what it is doing! Its anchor was the real thing - the [i]gradient[/i] of a stream flowing down sloping countryside.
Conversely, when I hopelessly tried A-Level Maths, I found 3-dimensional graphs easy because I could relate their abstract, algebraic ([i]x, y, z[/i]) co-ordinates to the Ordnance Survey maps' NGR and altitude points with which I was already familiar.
I've not had that lucky break with Integration though....
And as for Matrices..... Luckily I have never needed use these arcane abstractions! A scientist told me a typical use for them was in her work - solving gigantic blocks of simultaneous equations with 100 and more unknowns. The computer does the number-milling, of course, but she still needs understand the application and method. Not exactly everyday arithmetic for we lesser mortals trying to keep up with petrol prices or baking cakes!
The Maths I use now, rather than ordinary Arithmetic, is largely limited to simple geometry and mensuration, with trigonometry and pi occasionally.
'
Some with no head for numbers seem to take a perverse pride in it; almost boasting about being innumerate. I have no time for that. I accept many of us need know only simple weights, measures and money arithmetic. That's fair; but not a reason to sneer at Maths as a whole. Being no "Child Genius", or even just good at Maths, wrecked my dreams of becoming a professional scientist or engineer as both disciplines are highly mathematical. I did work in these, but at semi-skilled shop-assistant level.
Far from it. I was always a slow learner and Maths was one of my weakest subjects - as my post hints.
I understood logarithms eventually, some 30-40 years after leaving school, by a rather indirect route that gave a physical anchor to my thinking.
I had had to think of my own way around the problem - and if there is one thing most mathematics text-books dismally fail to do, it is to [i]explain[/i] the mathematics they illustrate.
Similarly with Calculus.
Although Calculus was taught in the Fifth Form of senior school, as part of the GCE O-Level Mthematics syllabus, I could not grasp Differentiation until a chance encounter in a geology club lecture probably thirty years later. Shown a numerical technique for analysing river gradients, something inspired me to write the simple formula in calculus notation. It dawned on me this was a sort of very basic [i]differentiation,[/i] and so finally twigged what that term actually means, hence what it is doing! Its anchor was the real thing - the [i]gradient[/i] of a stream flowing down sloping countryside.
Conversely, when I hopelessly tried A-Level Maths, I found 3-dimensional graphs easy because I could relate their abstract, algebraic ([i]x, y, z[/i]) co-ordinates to the Ordnance Survey maps' NGR and altitude points with which I was already familiar.
I've not had that lucky break with Integration though....
And as for Matrices..... Luckily I have never needed use these arcane abstractions! A scientist told me a typical use for them was in her work - solving gigantic blocks of simultaneous equations with 100 and more unknowns. The computer does the number-milling, of course, but she still needs understand the application and method. Not exactly everyday arithmetic for we lesser mortals trying to keep up with petrol prices or baking cakes!
The Maths I use now, rather than ordinary Arithmetic, is largely limited to simple geometry and mensuration, with trigonometry and pi occasionally.
'
Some with no head for numbers seem to take a perverse pride in it; almost boasting about being innumerate. I have no time for that. I accept many of us need know only simple weights, measures and money arithmetic. That's fair; but not a reason to sneer at Maths as a whole. Being no "Child Genius", or even just good at Maths, wrecked my dreams of becoming a professional scientist or engineer as both disciplines are highly mathematical. I did work in these, but at semi-skilled shop-assistant level.