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ArishMell · 70-79, M
Only 1 and 2, but I know of the uses for a few others.
Number 2, Napier's law of logarithms, is purely arithmetical, but part of a very powerful tool for easily solving awkward multiplication, division and power "sums". Although the calculator has displaced the log / antilog tables for arithmetic, logarithms are still important in many scientific laws. The most familiar perhaps is the deciBel, the logarithmic ratio used in measuring and calculating the intensities and powers of acoustic and electrical signals*.
Number 3 is part of the enormous field of Calculus used in so many Scientific and Engineering calculations.
Number 5 is used for solving problems involving square roots of negative values by what are called "Complex Numbers"; used for example in Vector calculations of a.c. electrical signals.
.
Most of the rest are parts of particular scientific areas - and anyway I regret to say that they are beyond any maths alone I could have ever learnt! Indeed, for most of those equations you need understand both mathematics and those specific fields to considerable depth.
Consequently most of them don't "do" anything on their own.
Number 1, Pythagoras' Theorem, only means anything as the proportions of a right-angled triangle. Standing alone, it is self-evident but lacks identity or purpose, because any number can be described by the sum of two squares, and any number has a square root.
Number 13, Einstein's equation so often quoted without any context, is not explicable without explaining the General Theory of Relativity, and the relationship the formula describes.
I don't know Fourier's original line of study but a mathematical tool called the Fast Fourier Transform is used in analysing the frequencies mixed up in complicated spectra.
' ' '
*(The deciBel and Bel are not single, linear units like the metre and gramme. Quoting a sound pressure-level as of [i]x[/i] dB, is shorthand within its understood context, for an otherwise rather unwieldy description.
For measuring sound in air, the full form is "[i]x[/i]dB referred to 20µPa". That 20 micro-Pascals is the faintest detectable by the fully-healthy human ear, so gives 0dB as the reference-level [i]but only for that purpose[/i]. Since the Bar (standard atmospheric pressure) = 100 000 000 000µPa, that shows just how staggeringly sensitive is the ear. )
Marine acoustics uses 1µPa for its 0dB reference.
Neither scale can use 0 micro-Pascals as its reference because that would give the arithmetically-impossible division by 0 in the conversion to deciBels . )
Number 2, Napier's law of logarithms, is purely arithmetical, but part of a very powerful tool for easily solving awkward multiplication, division and power "sums". Although the calculator has displaced the log / antilog tables for arithmetic, logarithms are still important in many scientific laws. The most familiar perhaps is the deciBel, the logarithmic ratio used in measuring and calculating the intensities and powers of acoustic and electrical signals*.
Number 3 is part of the enormous field of Calculus used in so many Scientific and Engineering calculations.
Number 5 is used for solving problems involving square roots of negative values by what are called "Complex Numbers"; used for example in Vector calculations of a.c. electrical signals.
.
Most of the rest are parts of particular scientific areas - and anyway I regret to say that they are beyond any maths alone I could have ever learnt! Indeed, for most of those equations you need understand both mathematics and those specific fields to considerable depth.
Consequently most of them don't "do" anything on their own.
Number 1, Pythagoras' Theorem, only means anything as the proportions of a right-angled triangle. Standing alone, it is self-evident but lacks identity or purpose, because any number can be described by the sum of two squares, and any number has a square root.
Number 13, Einstein's equation so often quoted without any context, is not explicable without explaining the General Theory of Relativity, and the relationship the formula describes.
I don't know Fourier's original line of study but a mathematical tool called the Fast Fourier Transform is used in analysing the frequencies mixed up in complicated spectra.
' ' '
*(The deciBel and Bel are not single, linear units like the metre and gramme. Quoting a sound pressure-level as of [i]x[/i] dB, is shorthand within its understood context, for an otherwise rather unwieldy description.
For measuring sound in air, the full form is "[i]x[/i]dB referred to 20µPa". That 20 micro-Pascals is the faintest detectable by the fully-healthy human ear, so gives 0dB as the reference-level [i]but only for that purpose[/i]. Since the Bar (standard atmospheric pressure) = 100 000 000 000µPa, that shows just how staggeringly sensitive is the ear. )
Marine acoustics uses 1µPa for its 0dB reference.
Neither scale can use 0 micro-Pascals as its reference because that would give the arithmetically-impossible division by 0 in the conversion to deciBels . )
ArishMell · 70-79, M
Sorry - it is not possible to type algebra and formulae correctly in SW's basic text-editor but I'll risk these copying properly from writing them in 'Word', and edit as necessary:
18. The Period, T, in seconds of a Simple Pendulum of length L.
[i]T = 2π √(L/g)[/i].
If not attributed directly to Galileo Galilei, it does formalise his observations.
.
19 Geometrical Progression from first term [i]a[/i] and common ratio [i]r[/i]; the [i]n[/i]th value =
[i]a.r^(n-1)[/i] NB: the index is on the ratio alone, not the product.
One practical application is in designing gear-boxes.
'
20 . Spherical-spreading attenuation of amplitude with distance D from source, of a radiated sound or other signal:
[i]Attenuation = 20Log(D)[/i] , in deciBels.
It is the familiar inverse-square law, but in logarithmic rather than linear terms. The logarithm is to base-10 but that is not possible to type properly here.
.
21. Conversion of Sound Pressure-Level from linear units to dB:
[i]SPL in dB = 20Log(measured / reference)[/i] , dB; where the measured and reference pressures are in µPa and again, base-10 logs.
The 0db level of the familiar noise-measuring deciBel scale is the lowest SPL discernable by the fully-healthy human ear. If you analyse human hearing in linear and dB terms, what you learn about the ear's sensitivity is awe-inspiring......
18. The Period, T, in seconds of a Simple Pendulum of length L.
[i]T = 2π √(L/g)[/i].
If not attributed directly to Galileo Galilei, it does formalise his observations.
.
19 Geometrical Progression from first term [i]a[/i] and common ratio [i]r[/i]; the [i]n[/i]th value =
[i]a.r^(n-1)[/i] NB: the index is on the ratio alone, not the product.
One practical application is in designing gear-boxes.
'
20 . Spherical-spreading attenuation of amplitude with distance D from source, of a radiated sound or other signal:
[i]Attenuation = 20Log(D)[/i] , in deciBels.
It is the familiar inverse-square law, but in logarithmic rather than linear terms. The logarithm is to base-10 but that is not possible to type properly here.
.
21. Conversion of Sound Pressure-Level from linear units to dB:
[i]SPL in dB = 20Log(measured / reference)[/i] , dB; where the measured and reference pressures are in µPa and again, base-10 logs.
The 0db level of the familiar noise-measuring deciBel scale is the lowest SPL discernable by the fully-healthy human ear. If you analyse human hearing in linear and dB terms, what you learn about the ear's sensitivity is awe-inspiring......
Sazzio · 31-35, M
@ArishMell Would you say we use these equation on a daily basis without realising? Like judging when to slow down a car (speed) when approaching (time) a narrow gap (size)? Ok it's just speed, time and length I know... but still?
Judging things by mental equations we never realise? Maybe we even use Algorithm, Pie spherical without knowing? 🤔
Judging things by mental equations we never realise? Maybe we even use Algorithm, Pie spherical without knowing? 🤔
ArishMell · 70-79, M
@Sazzio [i]Interesting question.....[/i]
Speed, time and distance are hardly "just", and yes, whatever our brains are doing to make the necessary judgements must be extremely subtle and complicated, far beyond the simple algebra of D = ST.
To a large extent we do this instinctively too, although we need learn to judge them for artificial contexts like driving a car.
I don't think we use the circular formulae and similar laws not directly applicable biologically, by instinct though. We need to learn them.
.
Your question ties in with my last 2 examples, relating to sound. I took an interest in wondering what bats might possibly image in their minds when hunting for insects in a wood at night; or navigating through a cave in which the darkness is absolute.
To help me I set out to understand sound and hearing at a basic level, using our own sense as an example because it is so easy to find texts for, anchored in our own experience. The biological principles are much the same.
What I realised is that in the little animal just a very few grammes of brain is carrying out some extraordinarily complex, rapid and responsive signal-processing to control its flying, breathing, calling and (hopefully for a hungry bat) prey-catching; while still leaving enough brain simply keeping the animal alive!
Further, if its roost is in a cave, the same physically-little brain has to deal with very peculiar acoustics without any help from sight, and remember them enough to find the way in and out safely.
The corollary is that of our brains, many, many times larger than a bat's; how much is busy with our primary animal physiology; compared to how much is handling our far greater speech, dexterity, intellectual and emotional capacities?
'
So if the human - or any animal - brain is "doing maths" instinctively and behind the scenes as it were, [i]just what sort of maths is it[/i]?
Speed, time and distance are hardly "just", and yes, whatever our brains are doing to make the necessary judgements must be extremely subtle and complicated, far beyond the simple algebra of D = ST.
To a large extent we do this instinctively too, although we need learn to judge them for artificial contexts like driving a car.
I don't think we use the circular formulae and similar laws not directly applicable biologically, by instinct though. We need to learn them.
.
Your question ties in with my last 2 examples, relating to sound. I took an interest in wondering what bats might possibly image in their minds when hunting for insects in a wood at night; or navigating through a cave in which the darkness is absolute.
To help me I set out to understand sound and hearing at a basic level, using our own sense as an example because it is so easy to find texts for, anchored in our own experience. The biological principles are much the same.
What I realised is that in the little animal just a very few grammes of brain is carrying out some extraordinarily complex, rapid and responsive signal-processing to control its flying, breathing, calling and (hopefully for a hungry bat) prey-catching; while still leaving enough brain simply keeping the animal alive!
Further, if its roost is in a cave, the same physically-little brain has to deal with very peculiar acoustics without any help from sight, and remember them enough to find the way in and out safely.
The corollary is that of our brains, many, many times larger than a bat's; how much is busy with our primary animal physiology; compared to how much is handling our far greater speech, dexterity, intellectual and emotional capacities?
'
So if the human - or any animal - brain is "doing maths" instinctively and behind the scenes as it were, [i]just what sort of maths is it[/i]?
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.
+++
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.
NerdyPotato · M
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Subsumedpat · 36-40, M
I would have to look most of those up to understand them, I use No. 1 the most though.
Indianboy · 31-35, M
None. Forumlas/equations are so dead boring :(
AngelKrish · 26-30, M
5 of them...i forgot physics...many things
Thatsright · 61-69, M
Five of them used to could.
likesnatural · 70-79, M
I think some people think too much!
chubbysteve99 · 51-55, MVIP
all but 9-10 and 15 to 17
Tracos · 51-55, M
ehm.... perhaps 1 ?? 😳
Jenny1234 · 51-55, F
None
Isthisit · F
None
ABCDEF7 · M
[u]1[/u]
0
0
lumberjackslam · 41-45, M
FeetAreFantastic · 41-45, M
NerdyPotato · M
@FeetAreFantastic samesies! ✋
FeetAreFantastic · 41-45, M
@NerdyPotato 🤝
Lilnonames · F