Dividing with Units: An Introduction to Math Problem Solving

[thedy]

Hi,I have found out,that I have problem to understand basic algebra.Especially dividing with units.I do not understand,just what is unit.For example:We have m^2/m=m.But what doe is it mean m^2/m?Dividing with units is easy.We have just some number,and split it to more pieces.But units is qualitative stuff.And qualitative stuff can not be splited to more pieces,can be?

And with this problem I have always complications to solve many and fundamental math problems from real life.

Or speed:d/t...how I have to imagine something in meters is splited by time...I know,that I can use analogy,that length is per time.But this cannot be always applied.

I observe,that this is my major problem...thanks for answers

 

[ShayanJ]

Multiplication and division for units shouldn't be thought of like those for numbers.The only interpretation is making new units using existing ones and nothing else!

 

[thedy]

Multiplication and division for units shouldn't be thought of like those for numbers.The only interpretation is making new units using existing ones and nothing else!

Thanks,but why then for units hold true mathematical properties like multiplication,or division?
m*m=m^2...What is it?Which analogy I can use from real life?
And what is unit then?
Or I can ask otherwise.Why l^3/l=l^2,or l^2/l=l,or l/l=1? l=length

 

[SteamKing]

If you have a rectangle which measures 4 m on one side and 5 m on the other, the area of the rectangle is
4 m * 5 m = 20 m^2, or 20 square meters.

Units like m/s can be thought of as rates. We express velocity in m/s, so a velocity of 20 m/s implies that one travels a distance of 20 meters each second.

I take it from your confusion you have never played with a ruler, used a stopwatch, or messed around with a balance scale.

 

[thedy]

If you have a rectangle which measures 4 m on one side and 5 m on the other, the area of the rectangle is
4 m * 5 m = 20 m^2, or 20 square meters.

Units like m/s can be thought of as rates. We express velocity in m/s, so a velocity of 20 m/s implies that one travels a distance of 20 meters each second.

I take it from your confusion you have never played with a ruler, used a stopwatch, or messed around with a balance scale.

Yes,but why 4 m * 5 m = 20 m^2?How did you invite,discover or define m^2?
And why new units are defined only by multiplication/division? What is unit?

 

[Mark44]

Yes,but why 4 m * 5 m = 20 m^2?How did you invite,discover or define m^2?

It might help to draw a picture using graph paper. If you draw a rectangle that is 4 cm long by 5 cm wide, how many 1cm X 1cm squares make up the rectangle? A square that is 1 cm by 1 cm has an area of 1 square centimeter, which is usually written as 1 cm².

And why new units are defined only by multiplication/division?

For the same reason that you can carry out a multiplication such as 2x² times 3xy to get 6x³y, but you can't make the sum 2x² + 3xy any simpler.

If you use a wrench to tighten a nut by applying a force of 10 lb using a lever arm of length 2 ft, you are applying 10 * 2 ft * lb (= 20 ft-lb) of torque to the nut.

It would make no sense whatsoever to add or subtract 10 lb and 2 ft.

What is unit?

A unit is some standard quantity or size or measure that pretty much everyone understands. For example, lengths can be measured in centimeters or inches (and many other units, such as miles, furlongs, meters, millimeters, nanometers, kilometers, parsecs, leagues, etc.). Each of these is a unit.

A unit could also measure area, volume, mass, weight, electric charge, voltage, frequency, and many others.

 

[thedy]

Thanks to all,but still I am confused.For example v=s/t.s/t is unit,but what i get if I divide displacement or length by time?Many ,,small" lengths? or somekind of hybrid small lengthtimes?
Thanks

 

[Mark44]

Thanks to all,but still I am confused.For example v=s/t.s/t is unit,but what i get if I divide displacement or length by time?Many ,,small" lengths? or somekind of hybrid small lengthtimes?
Thanks

If you drive a car for 120 miles at a constant speed, and it takes you two hours, your average speed is $\frac{120}{2}$ in units of $\frac{miles}{hours}$. In other words, your average speed is 60 mi/hr. The "mi/hr" unit is a ratio that tells you the number of miles per 1 hour.

You might be overthinking this... It's really not all that complicated.

If you have 6 apples and give them all to 3 friends equally, then they are getting 6/3 apples per person, or 2 apples/person.

The units mi/hr and apples/person represent ratios of how many miles are covered in one hour or how many apples are given to one person.

 

[thedy]

If you drive a car for 120 miles at a constant speed, and it takes you two hours, your average speed is $\frac{120}{2}$ in units of $\frac{miles}{hours}$. In other words, your average speed is 60 mi/hr. The "mi/hr" unit is a ratio that tells you the number of miles per 1 hour.

You might be overthinking this... It's really not all that complicated.

If you have 6 apples and give them all to 3 friends equally, then they are getting 6/3 apples per person, or 2 apples/person.

The units mi/hr and apples/person represent ratios of how many miles are covered in one hour or how many apples are given to one person.

Yes,I understand mi/hr ratio.But I am wondering,why scientists use only multiplication/division to create new units?By what is multiplication/division special?

 

[Windadct]

I agree with Mark - you may be over thinking or over complicating this. In most of the real math we do - that applies to something real - we have units, and the simplest ones are called Fundamental Units.

a Length x Length .. units of m^2 for example, will often be an AREA, while there some "units" for area, like an acre these are for convenience and not generally helpful for any scientific analysis. IMO it is always best to keep the FUNDAMENTAL UNITS - in mind - if not in your math.

For Speed - KPH - the P=per really the same thing as divided by... so the FUNDAMENTAL UNITS are Meters (Kilometers) and Time(Hours).

Displacement - a linear measurement, divided by time - is a Speed. If you divide a displacement ( length) by a length - you have removed the units - and if it is in your math as in solving some real world question -you PROBABLY made a mistake - but this case does come up some times and this results in a constant - there are some non-dimensional constants that are useful - but if you are asking the questions above, do not worry about these YET!

 

[Mark44]

Yes,I understand mi/hr ratio.But I am wondering,why scientists use only multiplication/division to create new units?By what is multiplication/division special?

See post #6.

 

[SteamKing]

Yes,I understand mi/hr ratio.But I am wondering,why scientists use only multiplication/division to create new units?By what is multiplication/division special?

Multiplication and division of units are special because addition and subtractions of units only makes sense when the units being added or subtracted are the same, for example, meters + meters or feet - feet.

For instance, if I measure a certain distance with my meter stick, I might come up with a distance of 5 meters. If I were to re-measure the same distance, but I used a meter stick to measure the first two meters, and then I switched to using a foot ruler to measure the remaining distance, I would come up with a measurement of 2 meters + 9 feet + 10 inches + a little bit left over. The distance is still 5 meters, but the second measurement uses a mixture of different units, which is very cumbersome to use in any further calculations unless they are reduced wholly to meters or wholly to feet.

 

[thedy]

Hi gentlemen,I am back with my questions.Yes,you may say,that I am overthinking once again :)
So,why do I have problem to imagine what is momentum=mv,or F=ma,or kilowatthour=kilowatt*hour,but I can imagine and explain intuitively what is m^2 or m^3?I checked probably all threads deling with units,what is multiplication and division,purpose of multiplication,why we can not add or substract two non-identical units and so on.But I have still problem in real life solve basic problems.
I am wondering how was discovered law F=ma,if there is no analogy.Like m^2,We can not see F like meter like a and acceleration like b.It means a*b=area.And if I do not know,what is F how I can count other problems with this law?
Thanks to all for any patient answers with way like Mr.Feynman.

 

[olivermsun]

Well many interesting concepts get discovered experimentally. You see something occurring with real objects, in the real world, and look for a way to describe them using known concepts and units.

For example, you pull something with a certain force and notice that it accelerates at a certain rate. Then you pull on two somethings (twice the mass) and it's harder by twice to accelerate (it accelerates at half the speed). So on and so forth.

We can't "see" force but we can feel it. We can notice acceleration because a certain object which is accelerating covers more and more distance over successive clicks of a clock.

 

[thedy]

Well many interesting concepts get discovered experimentally. You see something occurring with real objects, in the real world, and look for a way to describe them using known concepts and units.

For example, you pull something with a certain force and notice that it accelerates at a certain rate. Then you pull on two somethings (twice the mass) and it's harder by twice to accelerate (it accelerates at half the speed). So on and so forth.

We can't "see" force but we can feel it. We can notice acceleration because a certain object which is accelerating covers more and more distance over successive clicks of a clock.

Thanks I give another example:I noticed,that here on Forums many people advise to guys,that we have to notice if in some problem we have different units or the same.For example if we count something (add together) like F=m+a,we are certainly wrong.It seems from this like Newton unit was derived by this train of thought.Newton sat under the tree and was thinking,that we can not add two different units,so we must have F=ma,not F=m+a

Second example:As I said some problems we can really solve by knowing dimensions or units.Like for example we have 3 apples in one bag.So unit is 3 apples/1 bag and if we want to know how many apples we have in 2 bags we have to 3 apples/ 1 bag x 2 bag.So here I can see logic,I can imagine in my head bags and apples,visually I can imagine all these units,and I know what to do with them.I no even need to know some universal equation or function to solve this problem.Why is it not so with physical phenomena?Why in these trivial examples I know,when to add substract/divide multiply,but in physical laws I can not exaplain why is there multiplying sign and why not adding sign.

I am always listening that we can not mix apples with pears.Yes I understand this analogy,but because of mystery reason,this analogy I can not apply on F=ma or E=mc².

Even the s=v*t I can explain by similar like apples and baggs because we have unit s/t (like apples/bag) and if we want to know terminal length we must multiply by time,because we know length per one unit of time and therefore,how many time units we have,we know final length.So it seems like I can understand units and dimensions,multiplying and dividing only if in equation I have unit that is per something,because then we are finding some whole thing of something.So if we have some fraction and want to know the whole we must multiply by denominator(bag,time).

Thanks for reading my post.

 

[Stephen Tashi]

I am wondering how was discovered law F=ma,if there is no analogy.Like m^2,We can not see F like meter like a and acceleration like b.It means a*b=area.And if I do not know,what is F how I can count other problems with this law?

Some people may discover important physical laws by reasoning with units ("dimensional analysis"), but this is not the usual way they are discovered. The arithmetic of units is applicable to physical laws that are not "universal".

For example, suppose you have a complicated machine and you discover by experiment that the force F in Newtons that you exert on a lever is related to the distance the machine moves in meters by F = 3 X. The law F = 3 X is not a "universal" physical law. It only applies to your machine. Also, F = 3 X is incorrect for someone who measures distance in centimeters. To state the law so a person can apply it using any system of measurement, it is necessary to give the constant "3" the units Newton/meter. This informs people how to use other units to measure force and distance. The "meaning" of Newton/meter in the equation may be completely unclear, especially to someone who doesn't understand the complicated machine.

So-called "universal" physical laws are not simply equations. They only apply to special situations which must be described by words. When a person states a physical law as a mere equation, he assume his audience is sophisticated enough to understand the situation where it is applied. For example, F = MA does not mean that the force you exert on the lever of your toaster this morning is equal to the mass of your physics text times the acceleration of a car on the highway next week.

The use of physical units in equations can be viewed as a special case of a more general principle that a physical law should remain true under a change of coordinates. For example, changing the units of measure from meters to centimeters is an example of changing coordinates - even it is only involves marking off the "force axis" in different units.