my question is
why physical quantities need a unit?
Because that is the only way you can quantify things. You can say that Alice is taller than Bob, but without a unit, you can't make a difference between Alice being twice as tall as Bob to Alice being barely taller than Bob. (Even saying that Alice is twice as tall as Bob has introduce a unit of length, namely a "bob".)
Because they are expressed as a a number times something that is used as a reference and that has a dimension. You can't add meters and seconds because they have different dimensions. You can't add miles and kilometers without converting because they have a different reference
Ha, and: !
Jack is 12 away, but I'm 3 away from you, so you'll need to leave early in the day, say at 8, because it will take you longer to get to me than to Jack. If you travel at 65, you should be able to see all of us before 3.
Did any of that make any sense to you?
Your question is a good one. There's more to be had from this if you turn it around: some mathematical operations can only be performed on numbers (taking a logarithm, for example). So if you end up with something like Amplitude = ##\log##(length) you know that can't be right.
I think that question is a bit tautological. I would recommend a pragmatic approach to these things - unless you have a lot of free time to go at it in depth. A quantity is, by its nature, related to an amount of 'something', as has been mentioned above but we don't only use numbers with units. There are many numbers in nature that do not have units but we don't normally refer to them as Quantities. 'The Power Two' that we put in the Inverse Square Law could be looked upon as a Quantity but we don't actually call it a quantity as such. The Ratio of your height to mine is, arguably, a quantity and then you could argue that the 'units' of your height would be 'per Sophiecentur's Height".
Dimensional analysis is a powerful tool for checking that a formula 'could' be right when the units on one side balance those on the other side.
That reminds me of a tourist who visited St Andrews in Scotland (the home of golf) and asked a local how far it was to the beach. The local replied "och, about a drive and a good 6-iron"!
They don't call them woods any more ?
A "3-wood" is a different unit of length from a "drive". The number 1-wood is always called a driver.
I was visiting friends in South Carolina. I volunteered to go get more beer; she told me to go to the 7-11 store, "...it's down the road a piece, maybe a piece and a half..."
Theoreticians like to express a lot of quantities in terms of dimensionless constants, but this is really inconvenient in real experiments, or in trade or engineering. We need values we can quickly and accurately compare, and the standards of measurement are just units.
It is not enough to say the magnitude of a physical quantity is quite large, or quite small. We need to be able to compare the magnitudes of physical phenomena around us.
A unit of a physical quantity is simply the magnitude/size of that physical quantity that we humans have defined to amount to the number one, numerically.
Length is defined as the amount of space between any two points in space.
One metre was defined as the distance between some bar of platinum in France (if I recall correctly).
The standard of measure for length was the distance between the notches on this bar of platinum. It was changed to the distance travelled by light in a very short interval of time. We need to have a standard of measure so we can quantify the physical phenomena in terms of units.
If we can't measure it, we cant quantify it! and in order to measure it, we need to defined units for it.
In order to defined a unit for it, these physicists and smart people come together and say "Ok, this much of this physical phenomena amounts to ONE unit" and thus everything can now be quantified.
For example: Say we have a length of a piece of wood here:
We want to build something of it, we have a rough idea of how long it must be visually...
But because we dont have units, we don't know for sure if this is enough or not.
A physicist comes along and says one = sign is equal to ONE UNIT of wood. And thus we have units.
We take this distance of this equal sign and REPRODUCE it over rulers and other instruments. Now we can construct and measure things out in nature. The choice of the magnitude that represents one unit is also important, considerations must be given to:
1. It being easily reproducible.
2. Practical: what if one metre was defined to be as long as 1000 times the distance than what it is now? would that be practical for the purposes of construction, calculations, distance measurement?
I will speak of a situation where I had to invent my own units of measurement. I was choosing between two rooms to rent, and though both looked about the same size one seemed bigger. I did not have a ruler or any instrument of measure and the landlord only gave a rough estimate. So I decided to use my feet as the unit of measure. I walked across the length and width of both rooms counting the number of steps. Then I multiplied them to get a rough measure of their areas. Without a unit of measure, physical phenomena are meaningless to us if we want to:
1. measure them
2. Model them and their relationships mathematically
3. Use them to achieve and satisfy our practical needs in engineering and construction.
Jack, you and I would probably know exactly what was meant but only if we live on the same continent. Leaving out the units can be a useful data reduction strategy (or a simple code).
@abbas14 Suppose I told you that the temperature outside was 10. Would it mean anything to you? The units of temperature, among others are degrees C, degrees F, degrees K, and degrees R. So, which is it?
Suppose I told you that the energy is 100. Would it mean anything to you? The units of energy, among others are Joules, ergs, N-m, dyne-cm, electron volts, liter-atm., ft-lb, ##Pa-m^3##, calories, W-s, Horpower-hrs, and BTUs. So, which is it?
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