Complex dimensions and dimensional analysis
I am a bit confused as to where the 1/2 went in the final calculation of energy. shouldn't the final result be
M*L^2/2*T^2 ?
I am still not sure, after reading the material, what the answer to the question asked in the end of the text
is ? "Why can't we raise a quantity with one dimensionality, to the power of a quantity with another
dimensionality e.g., x^t ?" although we can multiply and divide quantities with different dimensionalities.
Are dimensionalites and units the same thing?
How will dimensionality change if we use functions like cosine, sine, and tangent
About the question posed at the end, I'm assuming this is because raising a quantity to a power is simply
multiplying it by itself many times, and it applies to whatever unit is attached (meters^2 is meters*meters),
but this can't be done with something like meters^seconds because that is a nonsensical unit?
In the article it states, "We can only add, subtract, or equate quantities that have the same
dimensionality", but when we solve for Gibbs free energy(delta G=delta H - TdeltaS), the equation
includes Temperature which is a different dimension?
What about dimensions that it doesn't really make sense to divide in real life or practicality such as light:
can you divide light or multiply light for example?
why can we mix different dimensions in multiplication and divisions but not for adding and subtracting?
If [X1]=L and [X2]=L why is it also true that [X1 + X2]=L ?
If...
[x1 + x2]=[x1]=[x2]=L, can we say that [x1] + L= [x2] and vice versa, [x2] + L= [x1]? Or is that not possible
because both x1 and x2, are in brackets, and you can only add up the same elements which have the
same dimension?
How would we define the restrictions of dimensionalities in respect to finding a derivative or an integral?
I have a pretty good understanding of this but my only question would be what do you mean when you
say magnitude of a quantity?
Would something like "L/T" constitute a new dimension altogether, or is it just a relationship between two
dimensions?
Aren't we putting a dimension to the power of another dimension with the equation P=e^(rt)? Why is that
equation allowed to be commonly used in math and biology?
Say you are equating x and y to L, so [x]=[y]=L - can you ignore any numerical values associated with X
and Y since you are only concerned with dimensions?
What are the rules for dimensionalities when the equation has both multiplication and division
Will there ever be a situation where the complex dimension involves both multiplication/division and
addition/subtraction? If so, how would we go about analyzing this dimension? Would we take an algebraic
approach?
For the equation for energy why does the dimensionality of [m] change to M?
So basically, any time there is a discrete number involved inside brackets such as [5v] it is just ignored?
So [5v]=[v]?
Why can we multiply and divide dimensions but not add them?
I know that we focus a lot on dimensions that we already know such as time, length, or velocity. But does
everything have a dimension that can be measured?
Are dimensions simply equating different values as part of the same measurement parameter? Is that
why added [x+y]=[y]=L even though the actual measurement may be different?
Do the same rules apply for measurements that are have complex dimensions such as
"meters/second^2"? What about constants?
Why can you divide and multiply quantities with different dimensionalities?
Is it possible to add two quantities together if they have different units?
How is [x1+x2]=[x1]=[x2]=L? Is the point of that just to say that their dimensions are the same? Because
they are not the same in magnitude; if so, I am not sure how.
Is there ever a situation where we can add measurements of different dimensions?
Is it possible to combine operations in dimensional analysis, such as combining addition and division?
If we can multiply and divide quantities of different dimensionalities, are we then able to incorporate them
into functions and take derivatives and integrals of them?
What is the difference between "quantity" and "magnitude"? I use these terms synonymously, so now I
struggle to understand the concepts that dimensions measure quantity NOT magnitude.
Are dimensions unitless? The reading says, "we can only add, subtract, or equate quantities that have the
same dimensionality." So if you have two dimensions of length, we can add them together. However,
what if one is in inches and one is in feet? Those have the same dimensionality, but not the same units.
So do dimensions not have units or do we just convert the units to match?
When there are 2 different dimensions in one equation, are you able to add/subtract to manipulate one of
the dimensions and not the other
Why is it that you can multiply and divide quantities of different dimensionality, but you can not add or
subtract them? How do you go about an equation that has all four processes (addition, subtraction,
multiplication, division) present within it?
Am I correct in thinking about a dimension as a math term that can only be described with measurement
(such as volume, area, length, width), and only variables with units that stem from the same core
(centimeters and millimeters both stem from meter, and therefore both lengths) as the ones we can add
or subtract? Would having an [x] that is measured in cm3 and an [f] measured in mL still be
added/subtracted since they are both in the dimension volume?
When a formula involves brackets, such as [x] = [y] = L, do the symbols in the brackets only reflect a type
of number, such as a length (I'm assuming this is a "dimension")? It's true that they don't refer to very
specific numbers, right?
Why does addition, or subtraction require quantities to have the same dimensionality when multiplication
can be done with quantities that have the same or different dimensionalities
Why can we only add, subtract, or equate quantities of the same dimensionality? Can we do more
complex mathematics like exponential or logarithmic relationships?
What does it mean for a number to have unit dimensionality?
When we change our choice of measurement scale, how does the quantity we are talking about change?
Is the use of the brackets the only change that needs to be made?
If you are solving an integral that, for example, involves the energy equation (1/2)m(v^2), how are the
units and dimensionality affected when you reach your answer?
Why are multiplication and division more straightforward in the sense that brackets are not required to
indicate dimensions, whereas in addition and subtraction they are absolutely necessary?
What is the proper notation for problems involving both addition/subtraction and multiplication/division?
The reading only provided separate examples but none where the equation required both.
On an x/y axis graph, are the x-values and y-values seen as having different dimensionalities because
they rely on different axes, even though they measure in the same units?
In the example about the equation for energy, E = 1/2mv^2, I just want to clarify that the reason the 1/2 is
dropped in the final equation (ML^2/T^2) is because it doesn't change even when the
measurements/dimensions change? Is that what it means to have unit dimensionality?
What can be done, if anything, to manipulate different dimensions in order to be able to add them?
As in the example in the reading, [1/2 m(v^2)] = [m(v^2)] = M(L/T)^2 = M(L^2)/(T^2), the 1/2 doesn't
change when any of our measurements change, so it has unit dimensionality, does this mean all
constants have unit dimensionality?
Why can you multiply and divide quantities with different dimenionalties but we can't add or subtract
quantities with different dimensionalities? Why doesn't adding/subtracting include that extra unit?
Since we cannot add or subtract quantities that have a different dimensionality, is it possible to express
one in terms of the other (unit) so that we could add or subtract them?
So in the end of everything, if you put brackets like this [x]=[y]=L and this could all be in inches and be
considered correct, but if you do x=y, this may not be equal numerically?
What is the purpose of writing an equation with strictly dimensions but no numerical quantities?
Why is a numerical value or unit dimensionality, for example the '1/2' in the quantity for energy [1/2 mv^2],
removed when simplifying the quantity?
What's the point of writing the certain equations in terms of its dimensions? As in, what's the benefit of
rewriting the velocity and energy equation as L/T and M(L)^2/(T)2?
when something is constantly growing or changing shape is there a certain way to measure the
dimensions of this object at a certain moment in time? How would you define the dimensions if it was
never the same as the minute before?
I understand that you cannot add/subtract different dimensions because the values are taken by different
kinds of measurements, but what changes in multiplication/division that allows us to overlook the different
dimensionality and relate them together?
Is the reason the 1/2 is taken away in the energy equation is because it is a constant?
Is it possible to add, subtract or equate quantities that have different dimensionality?
What exactly is unit dimensionality and does it have different properties than the other dimensionalities?
The reading states, ". . . dimensions only tell you about the type of a quantity and not its magnitude . . ." Is
there a way to incorporate quantity and magnitude into the same measurement?
In the Energy example of the text, why is it that the 1/2 in [1/2mv^2] is ignored when the equation is
simplified to [mv^2]. Is it because it is a constant? But then again isn't every variable in this equation
technically a constant?
Will be using brackets in equations or only to indicate symbols with the same dimensions?
How come we are allowed to use different dimensionalities for multiplication and division?
How does one correct for dimensionality when two values have different multiplied units of measure
(ft/sec x m/min)?
In the article, it is stated that we may multiply or divide the same or different dimensionalities to create
combinational quantities but may not add/subtract different dimensionalities. However, in the equation
v=vo+at, doesn't initial velocity (vo) and acceleration (a) have different dimensionalities? How do we
consider this equation valid?
It was stated that "We can multiply(or divide) quantities that have the same or different dimensionalities".
If this is true can it be done for all units, for instance, can we divide or multiply light by energy in different
amounts?
The reading says that different dimensions can not be added or subtracted, but a lot of different
dimensions can be derived from each other and wouldn't that be considered adding/ subtracting a
different dimension?
Units are immensely important, but why are we using bracket notation to know the dimensional of a
problem?
It doesn't ultimately solve the problem, it just tells me what the dimensions are, which can already be
identified by looking at them, all that's left is conversions. So why is bracket notation being so highly
stressed?
Similarly, we will be dealing with multi-dimensional units such as m/s or ft/s^2 or kg*m/s, so how does
bracket notation help for these kinds of problems, when couldn't we just leave it, in its original units?
After reading and giving thought about the article indicated above, How does dimensional analysis
become so much more challenging to complete when fluctuating between addition/subtraction and
multiplication/division?
With the idea that dimensionalities cannot be added or subtracted, yet can be multiplied or divided, can
we infer that on a graph, a final quantity can become 3-dimensional (ie. with three different planes of
axis)?
Is there no way to add or subtract quantities that have the same dimensionality? In the past, one of my
math teachers said otherwise but did not elaborate.
If we were given a value wouldn't we also be given the units so we could determine if the two numbers
are in the same dimension, therefore making the bracket notation unnecessary?
How do we know WHY we can't add and subtract different dimensionalities but we are able to multiply
and divide them? Is there a reason that we can do this? How is this applicable in math and physics? In
biology?
I'm still confused about dimensions and brackets. Can you use a real life analogy to explain it?
Can we multiply and divide different dimensions because equations, and the models created to represent
the relationships between the various factors, focuses on the change in one particular aspect of the
system- an aspect to which all factors influence? In the example of the energy calculations, are the
various dimensions multiplied because in the equation the main focus is the energy of the system rather
than the change occurring to the individual variables?
Why do we need to use the brackets to signify that we are talking about the units of Velocity, for example,
when the rule states that we can multiply and divide different dimensionalities? Isn't this kind of
redundant?
Will we have to use bracket notation when referring to dimensionality on assignments?
It mentions that you can multiply values that have different dimensionalities, thus creating new
dimensions. But since some dimensions can be both positive or negative, such as temperature, but other
can only be positive, such as length, does this create a situation where certain dimensions can not be
multiplied together?
Is the possibility of multiplying and dividing values with different dimensionality feasible because the units
are still independently expressed in the final mathematical product?
I do not have much to say about this one, it was all pretty straight forward. But in the section that spoke
about things we can do to two equations set equal to each other how would placing something that adds
to 0 be beneficial to solving the equation?
I am still confused about the brackets. Can you add brackets to, lets say a biology formula?
I'm confused about how [x1+x2]=[x1]=[x2]. Is it because we are only adding dimensions and not real
amounts/measurements?
I was reading the page under the "change" link and I was surprised that even in NASA there was issues
with engineers not using the same notations. How is it possible that such an intelligent group of people
haven't standardized this yet?
Using [x]=[y]=L and [t]=T, is x+y+t equal to L+T?
If we can multiply and divide different quantities, but we cannot add and subtract those different
quantities, does that mean we can only observe the relationship between different quantities using
multiplication and division?
One of the major restrictions is "We can only add, subtract, or equate quantities that have the same
dimensionality." However, if we try to add or subtract 2 quantities with the same dimensionality but
different magnitudes such as centimeters and millimeters, how can we compare the two?
Why are we allowed to ignore dimensionality of units for multiplication and division?
How do we handle the dimensionality of physical constants -- or those even considered when studying
the dimensionality of a physical expression?
How do these rules apply to calculating an Area? If you multiply the width and length, does that mean
they have different dimensionality or do they have the same dimensionality of length? Does Area have its
own dimensionality?
If we are only able to add quantities that are of the same dimensionality, then how is it that we can add
length and width together to find the perimeter of an enclosed area? Is it because they are measured in
the same units?
Why does the 1/2 disappear in the last equation energy: [1/2 mv^2] = [mv^2] = M(L/T)^2 = ML^2/T^2? If
the 1/2 does change when any of our measurements change, then how could [1/2 mv^2] = [mv^2] be
true?
What is the highest number of dimensions you can combine by multiplying or dividing (before it becomes
not useful anymore)?
What is an example of when we would use bracket notation?
Why would the sum of two quantities have to be in the same dimension but multiplying two different
dimensions is ok?
Why can we multiply/divide quantities with different dimensionalities, but can't add or subtract them?
Why can multiplication be used for two quantities with different dimensions but addition can't?
when using the brackets in these types of problem, like in the velocity example they just dropped the 1/2,
is this only when you are giving an example that is not specified to a situation?
At the end of this web page this question was asked "Why can't we raise a quantity with one
dimensionality, to the power of a quantity with another dimensionality?" I'm not really sure why this can't
be done as I have seen examples of this being done. For example, an equation for interest growth is
A=P(1+r)t where rate + 1 which has a one dimensionality is raised to the power t which has a different
dimensionality.
Being the reason that because dimensions only tell you about the type of a quantity can different physical
quantities have the same physical dimension?
Can you convert one dimension, into another, when you are adding or subtracting 2 different dimensions?
What does it mean by unit dimensionality?
If you multiply two quantities of the same dimensionality (ex: velocity), does that mean you can potentially
get a measurement with a new dimensionality?
Whenever we see a bracket around a variable, does that mean we only look at the dimension of the
variable, or can brackets mean different things in different situations?
When performing a dimensional analysis to equate units, in the case that there is a measurement with
two dimensions that are divided (such as velocity in m/s), how would that be solved? For example, how
would you equate speed to velocity? Could that be done?
Can you preform derivatives or integrals on dimensions? or is it just limited to +-/*?
Can we perform other mathematical functions on same or different dimensions? Such as natural log or
exponentiating dimensions
Why can we multiply or divide measurements of different dimensions but all other mathematical
operations cannot be applied? Is it because we can get a combined dimension such as velocity which is
distance over a time? Why then would power operations not be used?
I do not know the answer to the question at the end "Why can't we raise a quantity with one
dimensionality, to the power of a quantity with another dimensionality e.g., x^t ?" but to add to it, is it
possible to take the square root of a quantity e.g., 100cm^2 square root equals 10 cm?
I don't understand why is it that you can multiply and divide different dimensions, but you cannot add or
subtract even though you're still comparing two values?
When you do multiply or divide different dimensions do the new units count as entirely new dimensions as
well? For instance, if s=[x] and m=[y] does m/s=[z], a whole new dimension?
In physics, how do scientists determine which variables in a model are negligible when dealing with real
world systems? Couldn't many "negligible" variables together constitute a large amount of error when
dealing with an extremely complex system?
In the readings, it was mentioned that although you cannot add quantities with different dimensions, you
can multiply and divide quantities with different dimensions. Why is this useful and how do these new
values become meaningful when adding and subtracting them cannot be done?
Is it acceptable to use two variables in an equation that represent the same dimension but have wildly
different values (such as 10 and 300) to solve a problem?
What is we combine different dimensions but the units are all completely different and it is an extremely
complex combination of dimensions? What units will we use and do we get rid of any?
Will all constants have "unit dimensionality" such as the 1/2 in the example given on the web page?
Since the [1/2] in the energy equality did not change, would that apply to any variable just numbers?
So would that indicate that if you squared values, that would not be able to count unless you squared the
units, as presented in class
Would any scientist ever be able to come to the conclusion that the only way they can express a certain
unit is to use a + or - sign in between dimensions?
If only dimensionality matters when adding or subtracting, when does magnitude become important?
The way I understood the question, it seems that when there are brackets around a value, then the
number inside no longer matters. What the bracket does is it makes it so that you only look at the
dimension that the value is in, such as length or volume, rather than the actual units being used in the
equations. Is this why in class earlier you made it so that [x^2]=[B], where x=L and B=V?
How to units change when adding and subtracting things of similar / same dimensions. What about when
multiplying or dividing?
Why is it that in addition and subtraction, the dimensions must be the same, when in multiplication and
division, this isn't true?
Why are we only able to add or subtract quantities that have the same dimensionality but be able to
multiply and divide quantities of the same or different dimensionality?
What does it mean to have unit dimesionaltiy ?
Can you add or subtract quantities of different dimensionalities if you convert them to the same
dimensionality? Also is there an easier way to add or subtract these quantities?
Is it alright to express addition and subtraction of multiplied or divided combinations, for example, in this
way: [1/2mv^2 + mv^2]?
When we are looking at the equality for the energy, you mention that the 1/2 in the first equality listed
doesn't change when any of our measurements change. How can we be sure or how do we tell that this
value, or any value in any similar circumstance, does not change with the other measurements?