# Measuring Complex #'s

#### Swapnil

Yeah yeah, complex/imaginary numbers are as *real* as real numbers. Just like irrational numbers are every bit *rational* as rational numbers.

OK, so complex numbers are useful--very useful--but I think that a lot of people, including myself, still don't take them as a part of physical reality; they see them only as a mathematical abstraction. I mean, can you really make measurements in complex numbers? An ammeter can show that you have -1.333 amps flowing through your circuit, but it will never show that you have 8.2 - j3.2 amps flowing through your circuit.

Do you guys think that this uneasiness we feel using complex numbers would be better resolved if we would start using instruments that display measured quantities in complex numbers? Is such a thing even makes sense? It is possible? How much potential does this idea have?

I'd love to hear you guy's opinion on this and bring your perspectives on the table. Last edited:
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#### Gokul43201

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Ever used a digital lock-in amplifier? It displays real and imaginary parts of any signal with respect to a reference signal.

#### neutrino

Y
Do you guys think that this uneasiness we feel using complex numbers would be better resolved if we would start using instruments that display measured quantities in complex numbers? Is such a thing even makes sense? It is possible? How much potential does this idea have?
Even if such a machine existed, and as Gokul has pointed out that it does, the output is only going to be a pair of real numbers, because every complex number is an ordered pair of real numbers.

#### cesiumfrog

Worse, the output is a pair of rational numbers.. #### Hurkyl

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Do you guys think that this uneasiness we feel using complex numbers would be better resolved if we would start using instruments that display measured quantities in complex numbers? Is such a thing even makes sense? It is possible? How much potential does this idea have?
Most measuring devices do display quantities in complex numbers. It's just that they usually restrict themselves to the purely real subset of the complexes. #### rbj

an instrument that compared two sinusoids of the same frequency could display the "gain" of one sinusoid over the other as a complex number. but it's still an abstraction. i know others have objected to it, but i think that "imaginary" is an appropriate term for any number that, when squared, becomes negative. those numbers ain't "real", but we can imagine them. sorta.

#### Hurkyl

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Real numbers aren't any more real than imaginary numbers.

#### rbj

Real numbers aren't any more real than imaginary numbers.
that is a philosophical statement, not a scientific one, that i, as well as a lot of knowledgable people (some with PhDs) disagree with. and, even though i'm not a physicist (electrical engineer who does signal processing), i think i can hold my own with this, Hurk. you have all sorts of issues to get through, like defining/agreeing on what is real. i'm equating the concepts of "real numbers" to "quantities of physical stuff" and there is no case where some quantity of physical stuff (not an abstraction like $\Psi(x,y,z)$ ) is measured as an imaginary or complex number. fundamentally, when we measure or perceive the amount of something, it a real number. because of our limited precision, we also measure it as a rational number, but we know that, when limited precision is not an issue, that there are ideal ratios of quantity that are irrational numbers.

#### Hurkyl

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that is a philosophical statement, not a scientific one
Sure. And it was made in reply to a philosophical statement. :tongue:

Hurk. you have all sorts of issues to get through, like defining/agreeing on what is real.
Not necessarily -- the crucial part of my position operates on the formal level: that there is no scientific reason to give the real numbers a priviliged status. I think the crux of my position is probably best expressed via a parody of your argument:

I'm equating the concept of "3-dimensional real vector space" to "relative displacements between physical locations", and there is no case where a relative displacement is measured as an real number, or even a complex number. Fundamentally, when we measure or perceive a displacement between things, it is a vector.​

What you presented is the usual argument that the reals are somehow special -- but all it does is pass the buck: it makes no attempt to explain why "quantity of physical stuff" is more real than things like "position", "angle", "constellation point", "quantum state", or "gravitational field".

Our measuring devices measure in whatever mathematical structure they're built to measure. If we build a device that gives a complex number to something called "gain", then by golly, the result of that measurement is a complex number. If I have a protractor that measures in multiples of pi radians, then by golly, any measurement I make with it is an irrational number. Digital devices measure in rational numbers because it's convenient -- not because it's a fundamental physical restriction.

And we don't perceive things in numbers -- we perceive things in, for example, visual stimuli. Counting the objects we see is an abstract process we've trained ourselves to do. (In fact, identifying objects in our field of vision is a fairly abstract process too!) The only reason you don't think of it as such is because you learned that particular abstraction when you were very young. :tongue:

We use the real numbers to describe "quantities of physical stuff" because the real numbers are a mathematical structure that has the properties we would like to ascribe to "quantities of physical stuff". Not because the reals are somehow the only "real" number system.

The reason I care about this topic is a practical one -- all the alledgedly "abstract" mathematical structures are created precisely because they capture some specific, useful, and often quite concrete notion. Abstractophobia is, IMHO, very harmful to one's ability and education.

(Ironically, I would have thought an electrical engineer would be more likey than most to treat the reals and complexes as being on equal footing!)

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#### vanesch

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The reason I care about this topic is a practical one -- all the alledgedly "abstract" mathematical structures are created precisely because they capture some specific, useful, and often quite concrete notion. Abstractophobia is, IMHO, very harmful to one's ability and education.
Amen to that Just to see at what point the real numbers were perceived as bizarre, think of the shock the Pythagoreans underwent when they discovered that simple geometrical lengths could not be constructed using rationals alone...

#### rbj

Not necessarily -- the crucial part of my position operates on the formal level: that there is no scientific reason to give the real numbers a priviliged status.
dunno what you mean by "priviliged status". certainly numbers are a sorta "human" invention (but chimpanzees and parrots have been shown to be able to count and i think an intelligent alien life would come to know nearly the same mathematical concepts we do, in due time) but the concept of "quantity of stuff" trancends human invention. before humans were around to count, the quantities of physical stuff existed and the quantitative interaction (that we presently describe with physical law) between that physical stuff existed. all of that quantity corresponds to real numbers.

I think the crux of my position is probably best expressed via a parody of your argument:

I'm equating the concept of "3-dimensional real vector space" to "relative displacements between physical locations", and there is no case where a relative displacement is measured as an real number, or even a complex number. Fundamentally, when we measure or perceive a displacement between things, it is a vector.​

What you presented is the usual argument that the reals are somehow special -- but all it does is pass the buck: it makes no attempt to explain why "quantity of physical stuff" is more real than things like "position", "angle", "constellation point", "quantum state", or "gravitational field".
maybe i'm dense, but i don't see the connection in your parody.

anyway, i'm making no attempt to explain "why" about quantity of stuff, only the observation of physical stuff and how much. that "how much" is invariably a real number.

Our measuring devices measure in whatever mathematical structure they're built to measure. If we build a device that gives a complex number to something called "gain", then by golly, the result of that measurement is a complex number. If I have a protractor that measures in multiples of pi radians, then by golly, any measurement I make with it is an irrational number. Digital devices measure in rational numbers because it's convenient -- not because it's a fundamental physical restriction.
i'm not saying there's a fundamental physical restriction against irrational quantities, only imaginary quantities. no physical quantity is fundamentally measured as "imaginary". we can abstract description of physical situations by use of these mathematical constructs called "imaginary" or "complex", but the quantities that we really measure are real. and, because the finite precision of our means of measuring, the raw quantities are also rational, but we might convert the raw measured quantity to another (say, by multiplying by $\pi$ or $\sqrt{2}$ and that result can be irrational, but the irrationality of that is not meaningful because there is a raw quantity arbitrarily close to the one we measured that, when converted the same way, would turn out to be rational.

And we don't perceive things in numbers -- we perceive things in, for example, visual stimuli. Counting the objects we see is an abstract process we've trained ourselves to do. (In fact, identifying objects in our field of vision is a fairly abstract process too!) The only reason you don't think of it as such is because you learned that particular abstraction when you were very young. :tongue:
we perceive things quantitatively, even if only approximate. there was this tribe in the Amazon that had terms for "none", "one", "two", "three", or "many". even that crude calculus was quantitative to some extent. nonetheless physical quantity exists, whether we perceive it quantitatively or not.

We use the real numbers to describe "quantities of physical stuff" because the real numbers are a mathematical structure that has the properties we would like to ascribe to "quantities of physical stuff". Not because the reals are somehow the only "real" number system.
but that's exactly what makes them real. because quantities of real or actual physical stuff are measured, in terms of whatever unit (even natural units) as real numbers. real numbers are about quantities of stuff that is really there. that really exists.

The reason I care about this topic is a practical one -- all the alledgedly "abstract" mathematical structures are created precisely because they capture some specific, useful, and often quite concrete notion. Abstractophobia is, IMHO, very harmful to one's ability and education.
i am not advocating Abstractophobia. but i do subscribe to the Einsteinian notion that things should be described as simply as possible, but no simpler. abstraction can help simplify physics or signal processing or any esoteric discipline. complex numbers, matrices, metric spaces (hilbert) and functional analysis can help simplify conceptualizing some real system and i'm all for that. but i'm against forgetting what the real quantities were to begin with.

(Ironically, I would have thought an electrical engineer would be more likey than most to treat the reals and complexes as being on equal footing!)
i generally do. when i think of driving a linear, time-invariant system with a sinusoid, i don't think of

$$x(t) = A \cos(\omega t + \phi)$$

i think of

$$x(t) = \left( A e^{i \phi} \right) e^{i \omega t} = \mathbf{X} e^{i \omega t}$$

it makes my life much easier. but i remember that this is but one component of the real signal and that there is another implied component:

$$\left( A e^{-i \phi} \right) e^{-i \omega t} = \mathbf{X}^{*} e^{-i \omega t}$$

and i know that a linear, time-invariant system made with real components (there ain't any real resistors with an impedance of $(40 + i 30) \Omega$) will act on that complex conjugate component exactly as it did with the first complex compoent, except with all $+i$ replaced with $-i$ and the output will be exactly the same as the output of the first component except it will be conjugated. then the real or actual or true (whatever adjective you want) result will, again, be a quantity measured as a real number, if i were to put this on a scope or voltmeter or whatever instrument to measure it.

complex (and imaginary) numbers are useful in dealing with reality, but only if you remember that they are imaginary.

#### MeJennifer

Yeah yeah, complex/imaginary numbers are as *real* as real numbers. Just like irrational numbers are every bit *rational* as rational numbers.

OK, so complex numbers are useful--very useful--but I think that a lot of people, including myself, still don't take them as a part of physical reality; they see them only as a mathematical abstraction. I mean, can you really make measurements in complex numbers? An ammeter can show that you have -1.333 amps flowing through your circuit, but it will never show that you have 8.2 - j3.2 amps flowing through your circuit.

Do you guys think that this uneasiness we feel using complex numbers would be better resolved if we would start using instruments that display measured quantities in complex numbers? Is such a thing even makes sense? It is possible? How much potential does this idea have?

I'd love to hear you guy's opinion on this and bring your perspectives on the table. Since this is in the physics and not the philosophy section I like to point out that numbers are not measured neither real nor complex!
Numbers are simply mathematical abstractions.

In physics there is use for both real and complex numbers. And by the way real numbers are actually a subset of complex numbers.

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#### vanesch

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anyway, i'm making no attempt to explain "why" about quantity of stuff, only the observation of physical stuff and how much. that "how much" is invariably a real number.
Actually, not. "How much" is in fact always natural number. Any measurement is "a number of times a smallest unit of measurement". An ADC in a measurement system "measures" a number of times the smallest step in the voltage to be quantized. Of course, from this natural number of times a unit measurement, one can easily abstract away to a rational number of bigger units: A measurement can result in n times the unit, plus m times a tenth of the unit, plus k times a hundreth of a unit etc... So in a way, measurements can also be considered as rational numbers. But that's about it.

At no point you measure a REAL number. Real numbers are a mathematical abstraction, following from the closure of the rational numbers, and in fact most real numbers cannot even be written down in any way.

Once we are using abstractions of this kind, we can go on to other mathematical structures, such as vectors, complex numbers etc...

#### russ_watters

Mentor
The reason I care about this topic is a practical one -- all the alledgedly "abstract" mathematical structures are created precisely because they capture some specific, useful, and often quite concrete notion. Abstractophobia is, IMHO, very harmful to one's ability and education.
This topic comes up from time to time and it is just plain absurd. It really gets on my nerves. All any mathematical device is is a tool for describing something real. No one device is any more or less real than any other.

The term "imaginary number" is simply a victim of its own poorly chosen name.

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#### russ_watters

Mentor
dunno what you mean by "priviliged status". certainly numbers are a sorta "human" invention (but chimpanzees and parrots have been shown to be able to count and i think an intelligent alien life would come to know nearly the same mathematical concepts we do, in due time) but the concept of "quantity of stuff" trancends human invention. before humans were around to count, the quantities of physical stuff existed and the quantitative interaction (that we presently describe with physical law) between that physical stuff existed. all of that quantity corresponds to real numbers.
Is the square root of 2 any less real because you can't count it on your fingers? I mean seriously - you're suggesting that only things that monkeys can do with math count as real!

Btw, an alien we are likely to meet would certainly understand our system of math. It is for all intents and purposes a universal language.

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#### rbj

This topic comes up from time to time and it is just plain absurd. It really gets on my nerves. All any mathematical device is is a tool for describing something real.
that's not true. it might be said for "Applied Mathematics", but not always for the extremely abstract pure mathematics that these academics do.

The term "imaginary number" is simply a victim of its own poorly chosen name.
also quite debatable.

Is the square root of 2 any less real because you can't count it on your fingers? I mean seriously - you're suggesting that only things that monkeys can do with math count as real!
i didn't suggest that at all. i was only suggesting that the concept of "quantity" is not merely a human construct. other living beings have that concept, intelligent aliens have that concept, and, in fact, the concept applies to physical reality by use of quantitative physical law even if there weren't living beings around to contemplate or measure it. and with respect to physical reality, it's only the real quantities that apply even though we living beings might be about contemplate "imaginary numbers" as an abstraction.

Btw, an alien we are likely to meet would certainly understand our system of math. It is for all intents and purposes a universal language.
i would agree. they probably would have a concept of "complex numbers" with whatever name and they probably would not expect their voltmeters to measure a complex or imaginary quantity either.

Actually, not. "How much" is in fact always natural number. Any measurement is "a number of times a smallest unit of measurement". An ADC in a measurement system "measures" a number of times the smallest step in the voltage to be quantized. Of course, from this natural number of times a unit measurement, one can easily abstract away to a rational number of bigger units: A measurement can result in n times the unit, plus m times a tenth of the unit, plus k times a hundreth of a unit etc... So in a way, measurements can also be considered as rational numbers. But that's about it.
i completely agree with you, vanesch, but i do not see how that refutes anything i (as well as quite a few academics) have said about this.

this nature of metrology is something that i have been interested in a long time which is why i'm a proponent of Natural Units (Planck Units or a variant) for use in thinking about this stuff (even if they are impractical) and why i'm a big cheerleader for the likes of Michael Duff whenever he takes on a proponent of VSL or some other varying dimensionful universal "constant". whenever you talk about the amount of some dimensionful physical quantity, you have to also ask "with respect to what?".

At no point you measure a REAL number.
no, we measure quantities that are real and rational even if there is, conceivably, a physical quantity that is irrational (that we can only determine to a finite precision or as rational). but we don't measure physical quantities that are imaginary, because there aren't any.

there are physical quantities that are natural or whole numbers - these quantities are counted when measured. there are physical quantities that are rational and we might have a prayer to measure those exactly. there are even conceivable physical quantities that are irrational, like the distance between opposite corners of a perfect square of 1 unit length on a side - we can't measure it exactly but we can get closer and closer to the limit of precision of our ruler or tape-measure. there are even physical quantities that are intrinsically negative (or bi-polar). whether you assign "negative" to protons or electrons, one of them gotta be negative because you can add these physical quantities and get zero. so it makes sense to have a measure of charge that is intrinsically negative. but there are no physical quantities that we can measure that are imaginary or complex, only real. and while the result of our measurement must also be rational (due to inherent error or limitations of measurement), the quanitity being measured might, as far as we know, be irrational.

Real numbers are a mathematical abstraction, following from the closure of the rational numbers, and in fact most real numbers cannot even be written down in any way.
yes. there are an uncountably infinite number of real numbers that cannot be expressed exactly in any manner. all rational numbers can and some of these irrational numbers can have an abstract expression (like $\sqrt{2}$ or $\pi$) that has exact meaning even ef we can't get to an exact value)

Once we are using abstractions of this kind, we can go on to other mathematical structures, such as vectors, complex numbers etc...
fine. i agree with that also.

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#### Hurkyl

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no, we measure quantities that are real and rational
Here is a picture of my ruler. It measures in centimeters. It sure looks like I'm getting an irrational number when I use it to measure a length.

And you've already stated that measuring devices can measure complex quantities:

an instrument that compared two sinusoids of the same frequency could display the "gain" of one sinusoid over the other as a complex number.
The instrument measured gain. Gain is a physical quantity. Gain is complex. Ergo, the instrument measured a complex quantity.

(What is "gain" in this context? phase difference? Phase difference certainly isn't a real quantity)

Incidentally... Why do you think physical quantities even can be expressed as real numbers? One could develop all of classical mechanics based on, for example, the hyperreal numbers, and it would make all the same predictions. There is, classically, absolutely no way to tell if the universe has "real" lengths, or "hyperreal" lengths. (Or something entirely different)

Furthermore, why do you think physical quantity can only be expressed as a real number? The imaginary axis of the complexes is exactly as good as the real axis for, for example, measuring lengths. Here's a picture of another of my rulers: it measures in imcentimeters. (The conversion between centimeters and imcentimeters is that 1 cm = i imcm) yes. there are an uncountably infinite number of real numbers that cannot be expressed exactly in any manner.
Actually, I believe that statement cannot be proven. I don't think you can even prove that there exists a single real number that cannot be expressed exactly! There are some very subtle logical issues here -- for a bit of related flavor, see Skolem's paradox.

i'm a proponent of Natural Units
One important thing to note: for example, if there "exist", two incommensurate lengths (their ratio is not rational), then it is mathematically impossible to represent length as natural number multiples of some fundamental length.

#### vanesch

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Actually, I believe that statement cannot be proven. I don't think you can even prove that there exists a single real number that cannot be expressed exactly! Here I don't follow you. Given that you can only write a countable number of mathematical expressions (finite sequences of symbols), and given that the real axis contains a continuously uncountable number of reals, it should be obvious that not every real number has a formal expression to it, no ?

#### vanesch

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i completely agree with you, vanesch, but i do not see how that refutes anything i (as well as quite a few academics) have said about this.
What I meant, was: the only "quantities" we know about "for real" are measurements, which are essentially natural numbers: we count stuff. A first step of "modelisation" is to use rational numbers, but that's just a trick to do "unit conversion". However, the quantities we measure have only an indirect relationship with what might "be out there". It is not because, when we observe that a meter stick goes 3 times in the length of a pole, plus 6 times a tenth of the meter stick, plus 2 times a tenth of a tenth of a meter stick, that there is anything "real" about this number. It is just an observation we made.

Apart from that, we have abstract formal theories which are our "physical theories", which use more sophisticated mathematical constructions, and according to one's tastes, one takes some of these formal elements "for real". However, that's just an option, a hypothesis. In classical mechanics, we hence imagine that we live in some kind of 3-dim Euclidean space, with "points" in space. But that's on one hand just a mathematical formal tool, and the hypothesis of its reality is only that: a hypothesis.
You can now discuss exactly what variant of mathematically equivalent structures is "the real structure" of space: are it 3-tuples of real numbers, or are it points on a 3-dim affine space with distance (nothing to do with numbers!) or even something else ? But that's a discussion which looks a lot like finding out the number of angels dancing on the head of a pin.
As such, if complex numbers are part of a mathematical structure of a physical theory, it is entirely up to you to decide in what way this complex structure is "real out there" and in which way it is just a way to tie up two real numbers. All of this is hypothetical. What comes out of a good physical theory is a number which will, in the end, predict what will be the approximate rational ratio between "a meter stick" and a "measurement".

Personally, I tend to take the stance that it is very practical to accept the hypothesis that the formal elements of a physical theory are "real", with the caveat that this is in any case not anything more than a working hypothesis.

#### Swapnil

I am just curious, is it possible to normalize some fundamental physical constant to 1 so that all measurements would be rational? For example, if we set some fundamental physical constant like Planck's constant or something to 1, then is it possible that all the physical quantities we measure like electric field, gravitational force, etc would become would become rational?

Also, I think that the idea of devices not being able to measure certain things precisely or whatever is a rather tricky problem because the devices we use to measure the "physical reality" are not independent of "mathematical abstractions." I think we usually design devices under the assumption of some mathematical model.

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#### Hurkyl

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Gold Member Here I don't follow you. Given that you can only write a countable number of mathematical expressions (finite sequences of symbols), and given that the real axis contains a continuously uncountable number of reals, it should be obvious that not every real number has a formal expression to it, no ?
That's why it's a (pseudo)paradox. And I confess that I've been guilty of perpetrating that myth.

I pointed at Skolem's (pseudo)paradox -- he proved that there exists a model of set theory in which there are only countably many real numbers. (Despite the fact you can prove, in the model, that there are uncountably many real numbers) There's an internal vs external thing going on -- internally the naturals and reals can never be in bijection, but it's possible for an external bijectino to exist.

I can try and explain it in depth, but not right at this moment. (and this probably isn't the right thread for it anyways) Prompt me for it somehow if you want to hear it.

#### vanesch

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That's why it's a (pseudo)paradox. And I confess that I've been guilty of perpetrating that myth.

I pointed at Skolem's (pseudo)paradox -- he proved that there exists a model of set theory in which there are only countably many real numbers. (Despite the fact you can prove, in the model, that there are uncountably many real numbers) There's an internal vs external thing going on -- internally the naturals and reals can never be in bijection, but it's possible for an external bijectino to exist.

I can try and explain it in depth, but not right at this moment. (and this probably isn't the right thread for it anyways) Prompt me for it somehow if you want to hear it.
You have teached me something totally new I was absolutely not aware of Never heard of that. Must be a mathematician's secret #### Swapnil

I am just curious, is it possible to normalize some fundamental physical constant to 1 so that all measurements would be rational? For example, if we set some fundamental physical constant like Planck's constant or something to 1, then is it possible that all the physical quantities we measure like electric field, gravitational force, etc would become would become rational?

Also, I think that the idea of devices not being able to measure certain things precisely or whatever is a rather tricky problem because the devices we use to measure the "physical reality" are not independent of "mathematical abstractions." I think we usually design devices under the assumption of some mathematical model.

#### Crosson

Natural numbers 1,2,3... are "natural" because they can be used to count.

Rational numbers are rational because they represent parts of wholes; any sub set of a finite set is a fraction of the original set.

Real numbers are introduced because our geometric intuition suggest that objects such as right triangles exists, and we believe that lines can have any length, and not all such inuitively "real" elements of a geometry diagram are fractional parts of eachother.

The real numbers are the unique complete totally ordered field. The rationals are a totally ordered field (totally ordered captures their linearity, and calling them a field means they have all the familiar aspects of multiplication and addition). Completeness (not continuity) is the property of "having no holes" i.e. every convergent sequence of lines converges to something that is a line!
Viewed in this respect, the set of real numbers is perhaps too large a domain for physical measurements.

In summary, the reason people reject imaginary numbers is because they do not measure lengths, and all of our units are reduced to measurements of mass, length and time.

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