Fractions help.
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Fractions help.
I've been looking at going back to school to get a degree. I have been trying to reteach myself basic math so that I will be able to do the course work with less headache. I've been going over fractions and while I can memorize the steps I really would like to be able visualize and understand what's going on.
Like visualizing 4/3 ÷ 3/2 I could do the problem but but I'm having trouble understanding it in reference to a number line. I've seen a few videos but the all use examples were the denominators are the same.
Another question in the same vain is how a negative times a negative would look on a number line.
I know these are basic questions but I really want to understand and be able to see these things and not just follow steps to get an answer. Any help or direction is greatly appreciated.
Like visualizing 4/3 ÷ 3/2 I could do the problem but but I'm having trouble understanding it in reference to a number line. I've seen a few videos but the all use examples were the denominators are the same.
Another question in the same vain is how a negative times a negative would look on a number line.
I know these are basic questions but I really want to understand and be able to see these things and not just follow steps to get an answer. Any help or direction is greatly appreciated.
 doogly
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Re: Fractions help.
so we're set up a little weird using both kinds of notation for division, because that is what fractions are, right? 4/3 is four divided by three. So we say "dividing by 3/2" is the same as "multiplying by 2/3"  clearly these are inverses, we have 2/3 * 3/2 = 1, good times  so really we could say we're just rectifying the notation. It brings us to (4 * 2) / (3 * 3) and we get to start simplifying from there.
In reference to a number line? I'm not sure, that isn't really a point of reference for me for thinking about this kind of operation. Dividing just means "over," it just means multiplication by reciprocal. In general my thinking goes the other way now  if I see division, I read it is as a fraction, because fractions are what make the most sense. They have the pizzas, you know. So good.
For negatives, the number line is definitely not the right place to understand this, because negative is a little weird and meaningless there. You know multiplication gets you area, and if you have 5 in the x direction and 3 in the y direction, you have a rectangle of area 15 in the upper left quadrant, and why should that be negative? Well it shouldn't, areas don't get negative.
So let's say I have too many puppies. I'm trying to keep a balance sheet, how am I paying for these puppies. I lose 2 puppies this week, and they were costing me 20 bucks to maintain. Now I have a *profit* of 40 bucks.
Or, I have two electrons near each other. The coulomb force between them is given by F =  q_1 q_2 / r^2. Since these are both electrons, q_1 and q_2 are both negative numbers, and so the force is negative, which means inwards here. This is a bit more of a stretch but maybe it shows a glimpse of where this becomes important in a way that is less contrived than the puppy counting.
In reference to a number line? I'm not sure, that isn't really a point of reference for me for thinking about this kind of operation. Dividing just means "over," it just means multiplication by reciprocal. In general my thinking goes the other way now  if I see division, I read it is as a fraction, because fractions are what make the most sense. They have the pizzas, you know. So good.
For negatives, the number line is definitely not the right place to understand this, because negative is a little weird and meaningless there. You know multiplication gets you area, and if you have 5 in the x direction and 3 in the y direction, you have a rectangle of area 15 in the upper left quadrant, and why should that be negative? Well it shouldn't, areas don't get negative.
So let's say I have too many puppies. I'm trying to keep a balance sheet, how am I paying for these puppies. I lose 2 puppies this week, and they were costing me 20 bucks to maintain. Now I have a *profit* of 40 bucks.
Or, I have two electrons near each other. The coulomb force between them is given by F =  q_1 q_2 / r^2. Since these are both electrons, q_1 and q_2 are both negative numbers, and so the force is negative, which means inwards here. This is a bit more of a stretch but maybe it shows a glimpse of where this becomes important in a way that is less contrived than the puppy counting.
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Re: Fractions help.
Note: I'm not a teacher. By far.
If you consider a negative number as "n steps to the left" and multiplying with a positive number as "repeat n times", then multiplying with a negative number becomes "undo n times". And undoing steps to the left naturally means taking steps to the right. Although, this perspective considers the left and the right of the multiplication sign as different, while the beauty of adding and multiplying is that the left and right side can be swapped at will.
To be only halfjoking: it makes sense geometrically if you jump to the complex numbers [insert link to a good source for complex multiplication with geometry]. A complex number can be described with a magnitude (that can't be negative) and a direction (counterclockwise from the xaxis), with all the positive numbers at an angle of 0° and all negative numbers at an angle of 180°. When multiplying you multiply the magnitudes as usual and you add the angles. So 5*8 (or (5, 180°) times (8, 180°)) has a magnitude of 5*8=40 and an angle of 180°+180°=360° which is the same as 0°, which means the number is on the positive side.
In any case, from what I recall from my childhood, the understanding developed after just following the steps loads and loads of times. (And then at university I learned the words/math to describe that understanding.)
Fireinthemarsh wrote:Another question in the same vain is how a negative times a negative would look on a number line.
If you consider a negative number as "n steps to the left" and multiplying with a positive number as "repeat n times", then multiplying with a negative number becomes "undo n times". And undoing steps to the left naturally means taking steps to the right. Although, this perspective considers the left and the right of the multiplication sign as different, while the beauty of adding and multiplying is that the left and right side can be swapped at will.
To be only halfjoking: it makes sense geometrically if you jump to the complex numbers [insert link to a good source for complex multiplication with geometry]. A complex number can be described with a magnitude (that can't be negative) and a direction (counterclockwise from the xaxis), with all the positive numbers at an angle of 0° and all negative numbers at an angle of 180°. When multiplying you multiply the magnitudes as usual and you add the angles. So 5*8 (or (5, 180°) times (8, 180°)) has a magnitude of 5*8=40 and an angle of 180°+180°=360° which is the same as 0°, which means the number is on the positive side.
In any case, from what I recall from my childhood, the understanding developed after just following the steps loads and loads of times. (And then at university I learned the words/math to describe that understanding.)
Re: Fractions help.
Fireinthemarsh wrote:Like visualizing 4/3 ÷ 3/2 I could do the problem but but I'm having trouble understanding it in reference to a number line. I've seen a few videos but the all use examples were the denominators are the same.
You can make the denominators the same. A fraction stays the same value if you multiply denominator and numerator by the same number.
So 4/3 = (2*4)/(2*3) = 8/6
and 3/2 = (3*3)/(3*2) = 9/6
This means that the ratio of the two fractions is 8/9.
Fireinthemarsh wrote:Another question in the same vain is how a negative times a negative would look on a number line.
Instead of a number line, I think it is better to look at a multiplication table.
Code: Select all
: 1 2 3

1: 1 2 3
2: 2 4 6
3: 3 6 9
You can extend the table to the left and upwards to include cases where one of the factors is negative.
Code: Select all
: 3 2 1 0 1 2 3

3: . . . 0 3 6 9
2: . . . 0 2 4 6
1: . . . 0 1 2 3
0: 0 0 0 0 0 0 0
1: 3 2 1 0 1 2 3
2: 6 4 2 0 2 4 6
3: 9 6 3 0 3 6 9
There is an obvious pattern to each row, and similarly to any column. How would you extend that pattern to fill in the dots in the top left corner? This tells you what negative times negative should be.
You can also prove mathematically that negative*negative=positive if you want numbers to have some nice properties. Those properties are:
 that multiplication of any number by zero gives zero. (x*0=0, 0*x=0 for any number x)
 a number plus its negation equals zero. (x + (x)=0 for any number x)
 the distributive law. (i.e.that x*(y+z) = x*y + x*z, for any numbers x,y,z, and similarly (x+y)*z = x*z + y*z )
The proof is this: For any two numbers a and b, we have
a*b =
a*b + 0 =
a*b + (a)*0 =
a*b + (a)*(b+b) =
a*b + (a)*b + (a)*(b) =
(a+a)*b + (a)*(b) =
0*b + (a)*(b) =
0 + (a)*(b) =
(a)*(b)
 Eebster the Great
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Re: Fractions help.
These responses should make the formal math pretty clear, but I'll try to give more intuition. One way to view division is finding the height of a rectangle whose area is the dividend and whose width is the divisor. For instance, 12 / 3 can be viewed as finding the length of a rectangle with width 3 and area 12. The idea is that a unit square has area 1, so a rectangle with unit width (width=1) has area equal to its length (area = length * width). So if you have a rectangle and you know its area is 12, you can literally divide it into three pieces, call the width of each 1, and see what the height is. That's the same as taking a rectangular array of beans and dividing them up into equal groups and counting the number in each.
So dividing by a whole number is clear, but to divide by a fraction takes a bit of thought. When you divide by three, you are making three rectangles with the same total width as the original and seeing how long each is. If you divide by a third, you are making a third of a big rectangle as wide as the original and seeing how long that big rectangle would need to be. In other words, 4/(1/3) asks for the length of a rectangle with width 1/3. A rectangle of width 1 and height 12 has area 12, so a rectangle with just a third its width would need triple the height (36) to have the same area. Thus dividing by a third is no different from tripling. And in general, dividing by 1/n is equivalent to multiplying by n.
In terms of the number line, recall that multiplying and dividing are inverse operations. If you multiply by x and then divide by x, you get back what you started with. So if we think of multiplying by 2 as mapping every point on the number line to a new point, we see everything getting twice as far from zero. So to undo that by dividing by 2, we have to map every point to one half as far from zero. This is by definition multiplying by one half.
One way to think of multiplying and dividing by other fractions (when the numerator isn't 1) is in two steps: first multiply or divide by the numerator, then the opposite for the denominator. But more important is to keep in mind the place on the number line. Regardless of exactly what 6 / (30/11) is, clearly 30/11 is close to 30/10, which is equal to 3. So the whole thing should be about 6/3 = 2. Moreover, we know 30/11 is actually a bit less than 3, so 6/(30/11) is a bit more than 2. This type of thing is easier to do with decimals by far, which is one of their chief advantages, but a bit of work with fractions makes it pretty straightforward there too.
Mathologer has a good video about building intuition for why multiplying two negative numbers gives a positive number.
So dividing by a whole number is clear, but to divide by a fraction takes a bit of thought. When you divide by three, you are making three rectangles with the same total width as the original and seeing how long each is. If you divide by a third, you are making a third of a big rectangle as wide as the original and seeing how long that big rectangle would need to be. In other words, 4/(1/3) asks for the length of a rectangle with width 1/3. A rectangle of width 1 and height 12 has area 12, so a rectangle with just a third its width would need triple the height (36) to have the same area. Thus dividing by a third is no different from tripling. And in general, dividing by 1/n is equivalent to multiplying by n.
In terms of the number line, recall that multiplying and dividing are inverse operations. If you multiply by x and then divide by x, you get back what you started with. So if we think of multiplying by 2 as mapping every point on the number line to a new point, we see everything getting twice as far from zero. So to undo that by dividing by 2, we have to map every point to one half as far from zero. This is by definition multiplying by one half.
One way to think of multiplying and dividing by other fractions (when the numerator isn't 1) is in two steps: first multiply or divide by the numerator, then the opposite for the denominator. But more important is to keep in mind the place on the number line. Regardless of exactly what 6 / (30/11) is, clearly 30/11 is close to 30/10, which is equal to 3. So the whole thing should be about 6/3 = 2. Moreover, we know 30/11 is actually a bit less than 3, so 6/(30/11) is a bit more than 2. This type of thing is easier to do with decimals by far, which is one of their chief advantages, but a bit of work with fractions makes it pretty straightforward there too.
Mathologer has a good video about building intuition for why multiplying two negative numbers gives a positive number.
 ThirdParty
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Re: Fractions help.
I think Jaap and Eebster are right to encourage you to think of multiplication and division as happening in two dimensions rather than one. That'll be necessary when you start multiplying and dividing things that aren't numbers: meters, grams, unknown variables, etc. But you asked how to visualize it on the number line, so I'll give you the onedimensional interpretation.
Let's start by establishing some notational conventions for representing addition and subtraction on a number line. We'll interpret "2+1" as meaning "start at 2 and move 1 step forward". "21" means "start at 2 and move 1 step backward". I'll mark the starting point with an "!" and the ending point with a "":
What if the addend or subtrahend is negative? Well, adding a negative number takes you backward, while subtracting a negative number takes you forward:
Next comes multiplication. "2x3" means "start at 0 and add 3, 2 times".
What would it mean to do something a negative number of times? Well, like Flumble said above, since "2x3" meant "start at 0 and do +3, 2 times", "(2)x3" means "start at 0 and undo +3, 2 times". And the way you undo addition is by subtraction.
To represent division on the onedimensional number line, we can interpret it as partition: "6÷2" will mean "divide 6 into 2 equal parts, and then find the size of the first part":
To use fractions, we need to partition the number line itself so that there are multiple steps between each whole number:
Finally, we need a way to represent things that happen a nonwhole number of times:
Now we can do the problem you asked about, (4/3)÷(3/2)=(8/9):
Fireinthemarsh wrote:how a negative times a negative would look on a number line
Let's start by establishing some notational conventions for representing addition and subtraction on a number line. We'll interpret "2+1" as meaning "start at 2 and move 1 step forward". "21" means "start at 2 and move 1 step backward". I'll mark the starting point with an "!" and the ending point with a "":
Code: Select all
number line: 10+1+2+3+
2: 
2+1=3: !+
31=2: !
1+2=3: !+.+
32=1: .!
(1): 
01=(1): !
23=(1): ..!
(1)+3=2: !+.+.+
What if the addend or subtrahend is negative? Well, adding a negative number takes you backward, while subtracting a negative number takes you forward:
Code: Select all
number line: 210+1+2+3+
(2)+3=1: !+.+.+
3+(2)=32=1: .!
1(2)=1+2=3: !+.+
(1)(3)=(1)+3=2: !+.+.+
Next comes multiplication. "2x3" means "start at 0 and add 3, 2 times".
Code: Select all
number line: 6543210+1+2+3+4+5+6+
0+3=3: !+.+.+
3+3=6: !+.+.+
2x3=0+3+3=6: !+.+.+!+.+.+
3x2=0+2+2+2=6: !+.+!+.+!+.+
2x(3)=0+(3)+(3)=(6): ..!..!
What would it mean to do something a negative number of times? Well, like Flumble said above, since "2x3" meant "start at 0 and do +3, 2 times", "(2)x3" means "start at 0 and undo +3, 2 times". And the way you undo addition is by subtraction.
Code: Select all
number line: 6543210+1+2+3+4+5+6+
3x(2)=0+(2)+(2)+(2)=(6): .!.!.!
(2)x3=033=(6): ..!..!
(2)x(3)=0(3)(3)=0+3+3=6: !+.+.+!+.+.+
Fireinthemarsh wrote:visualizing 4/3 ÷ 3/2 I could do the problem but but I'm having trouble understanding it in reference to a number line
To represent division on the onedimensional number line, we can interpret it as partition: "6÷2" will mean "divide 6 into 2 equal parts, and then find the size of the first part":
Code: Select all
number line: 0+1+2+3+4+5+6+
2x3=6: !+.+.+!+.+.+
3x2=6: !+.+!+.+!+.+
6÷2=3: (÷.÷.÷÷.÷.÷)
6÷3=2: (÷.÷÷.÷!÷.÷)
To use fractions, we need to partition the number line itself so that there are multiple steps between each whole number:
Code: Select all
number line: 0+.+.+.+.+.+1+.+.+.+.+.+2+
1+1=2: !+.+.+.+.+.+
(1/2): 
1÷2=(1/2): (÷.÷.÷÷.÷.÷)
(1/3): 
2x(1/3)=(2/3): !+.+!+.+
(2/3)÷2=(1/3): (÷.÷÷.÷)
2÷3=(2/3): (÷.÷.÷.÷÷.÷.÷.÷!÷.÷.÷.÷)
(1/2)+(2/3)=(7/6): !+.+.+.+
(2/3)+(1/2)=(7/6): !+.+.+
Finally, we need a way to represent things that happen a nonwhole number of times:
Code: Select all
number line: 0+.+1+.+2+.+3+.+4+.+5+.+6+
5x(1/2)=(5/2): !+!+!+!+!+
(1/2)x5=5/2: !+.+.+.+.+ . . . . .
2x(5/2)=5: !+.+.+.+.+!+.+.+.+.+
5÷2=(5/2): (÷.÷.÷.÷.÷÷.÷.÷.÷.÷)
2+(1/2)=(5/2): !+
(2+1/2)x2=5: !+.+.+.+!+.+.+.+!+.+ . .
5÷(2+1/2)=2: (÷.÷.÷.÷÷.÷.÷.÷!%.%) . !
Now we can do the problem you asked about, (4/3)÷(3/2)=(8/9):
Code: Select all
number line: 0+.+.+.+.+.+.+.+.+1+.+.+.+.+.+.+.+.+2+.+.+.+.+.+.+.+.+3+
4x(1/3)=(4/3): !+.+.+!+.+.+!+.+.+!+.+.+
8x(1/9)=(8/9): !+!+!+!+!+!+!+!+
(4/3)÷(1+1/2)=(8/9): (÷.÷.÷.÷.÷.÷.÷.÷%.%.%.%) . . . !
(3/2)x(8/9)=(4/3): !+.+.+.+.+.+.+.+!+.+.+.+ . . . .
 gmalivuk
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Re: Fractions help.
Fireinthemarsh wrote:Another question in the same vain is how a negative times a negative would look on a number line.
I agree with doogly that a number line is probably simply not the way to visualize negative multiplication.
I still remember the way I internalized this back when I was first learning negatives, which is in terms of money. If you have 5 dollars, it means you owe 5 dollars. One way you can owe $5 is if you have a bill for $5 that you need to pay. (And one way you can have $5 is if you have a check for $5 that you need to deposit.)
positive x positive = you have 3 checks of $5 each, so you have $15 (or will once you deposit them)
positive x negative = you have 3 bills of $5 each, so you owe $15 total
negative x positive = you "owe" 3 checks of $5 each, so you owe $15 total (owing checks means you have to send them out to other people, for example)
negative x negative = you "owe" 3 bills of $5 each, so you have $15 (or will once those people pay the bills)
 Eebster the Great
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Re: Fractions help.
It might be true that number lines are not very useful for understanding negative multiplication, but I feel like answering the implied question anyway.
Multiplying really does two things on the number line, not just one. Multiplying by a real number scales by its magnitude and reflects by its sign. What I mean is that if you have a point at, say, 3, and you multiply it by 2, first you scale the number up by a factor of 2 (so you get 6), then you reflect it about zero (so you get 6). If you multiply a negative number by a negative number, you again must scale and reflect, which will bring it back to the positive side. Changing sign (reflecting about the origin) is just an essential part of what multiplying by a negative number means.
By the way, if you multiply by a complex number (which has a real part and an imaginary part), you still scale up by the magnitude, and then you rotate by the argument (complex angle). It turns out that negative real numbers have a complex argument of 180 degrees, so you are rotating the points halfway around the plane, putting them back on the opposite side of the real line.
Multiplying really does two things on the number line, not just one. Multiplying by a real number scales by its magnitude and reflects by its sign. What I mean is that if you have a point at, say, 3, and you multiply it by 2, first you scale the number up by a factor of 2 (so you get 6), then you reflect it about zero (so you get 6). If you multiply a negative number by a negative number, you again must scale and reflect, which will bring it back to the positive side. Changing sign (reflecting about the origin) is just an essential part of what multiplying by a negative number means.
By the way, if you multiply by a complex number (which has a real part and an imaginary part), you still scale up by the magnitude, and then you rotate by the argument (complex angle). It turns out that negative real numbers have a complex argument of 180 degrees, so you are rotating the points halfway around the plane, putting them back on the opposite side of the real line.
 Soupspoon
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Re: Fractions help.
(Readers in some parts of the world may consider bills (banknotes) to be positive wealth and checks (restaurant customer 'invoices') to be negative. They'd maybe pay their checks with bills, whilst I might pay a bill with a cheque (though rarely, these days). But that's not a math(s) thing. You got the maths thing ok. )
 Eebster the Great
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Re: Fractions help.
TIL all meanings of the English word "check" derive from terms medieval Frenchmen used in reference to or while playing chess, and thus ultimately from the Persian word "shah." I guess if a hockey player checks me, he is threatening to capture me. And if I check your coat, I am restricting the allowable moves of coat thieves to only those which would avoid my threat of exposing their attempted theft by revealing their lack of a receipt. Something like that.
 Xanthir
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Re: Fractions help.
Eebster: citation for that? I'm really curious about this!
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))
 Eebster the Great
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Re: Fractions help.
Xanthir wrote:Eebster: citation for that? I'm really curious about this!
Etymonline
Re: Fractions help.
This thread reminds me of a time my girlfriend and I were watching a Ghibli movie wherein the young protagonist was struggling to understand dividing by fractions in terms of cutting up apples into pieces, and my girlfriend didn't understand how the character couldn't understand simple things like how "dividing one apple by a half is just a half apple" and why is this character making it so hard on herself, indicating that she (my girlfriend) also didn't understand the nature of the problem. I was eventually able to explain why the character's having a hard time with it, and what the nature of the problem was, by illustrating how it's kind of a weird question to ask in the context of cutting up apples: asking what is an apple divided by a half is asking how many halfapples it takes to make one apple, but we don't normally build apples out of fractional pieces thereof IRL.
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"I am Sam. Sam I am. I do not like trolls, flames, or spam."
The Codex Quaerendae (my philosophy)  The Chronicles of Quelouva (my fiction)
Re: Fractions help.
"And we put it all together into one piecespart!"Pfhorrest wrote:but we don't normally build apples out of fractional pieces thereof IRL.
(Points to anyone who knows where that came from. Internets to whoever can link to the original!)
Jose
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