Avoiding division absurdities.

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In summary, the conversation discusses the concept of .9r and its relation to the number 1. The main points include the use of order of operations in mathematics, the equation x=.9r, and the belief that .9r is equal to 1. The conversation also touches on the controversy surrounding this concept and the importance of understanding the real numbers in mathematics.
  • #1
boit
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How do people end with .9r?
Apart from trying to derive square root of numbers using Newton's method, the biggest culprit is how we employ the authorised order of operation in mathematics i.e. BODMAS, PENDMAS e.t.c. Specificaly where we place the quantities.
I saw an equation being used to 'reduce' .9r into 1 (or is it to collapse it into 1?) that went thus
x=.9r
10x=9.9r
subtracting the first equation from the second and we get
9x=9
dividing each side by 9 we find x=1 proving .9r is actually (equal to) 1.
Here is the problem. If we decide to start with division before multiplication, we end up in the same hole that we're trying to get ourselves out of.
Check this. (1/9)x9=?
Now let's divide 1 by 9, what do we end up with? .111 . . . .i.e. 0.1r
multiply that by 9 and we end up with .9r
If we could only have re-arranged the equation to be 1x(9/9)
That way we will avoid unnecessary controversies (but again just like in tabloids, maths seems to thrive in controversies).
 
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  • #2
I think what you are trying to say is:

##1 = \frac{9}{9} = \frac{1 \times 9}{9} = \frac{1}{9} \times 9 = 0.\overline{1} \times 9 = 0.\overline{9}##

Of course when students see this they ask "but 1 divided by 9 isn't really 0.1 recurring, it's just an approximation" and we are back to square one.

That way we will avoid unnecessary controversies (but again just like in tabloids, maths seems to thrive in controversies).

I don't know what you are saying here. Students need a proper understanding of the reals. Just look at this thread, especially from post #34 onwards. As Hurkly wrote in that thread
Did you know there are a number of people who think numbers aren't "exact"? That people get the idea that there is no such number as "one third", with reasoning that decimal can only provide an approximate value of such a thing? Non-terminating decimals really need some amount of coverage, to properly connect decimal notation to arithmetic and algebraic knowledge.
Your solution doesn't address this, and suffers the same problems that the original solution you posted suffers from.
 
  • #3
First of all. .9999... is = to 1.

Second of all, these 'absurdities' like PEMDAS are crucial to mathematics. There are the rules math follows.

To say they are arbitrary and wrong is like saying the dictionary definition for the word 'the' is wrong because 'the' can mean anything you want. Sure, you're right in a sense. But you're right in a useless way which gets you no where.

We define an order of operations so that our math works. Without it, math can do nothing.

Remember, all of math is made up, so we can define whatever we want (as long as the system is consistent). It's just a testament to the power of the things we 'define arbitrarily' that we can use math to so accurately describe the real world.
 
  • #4
Vorde said:
Second of all, these 'absurdities' like PEMDAS are crucial to mathematics. There are the rules math follows.

Hardly. Order of operations is "needed" only because frankly we can't be bothered writing the parenthesis ##(3 \times 4) + 5## every time we have more than two operations in an expression. It's a notational convention, and has nothing to do with the underlying mathematics. The only part that is "crucial" is the parenthesis.

Just look at any non-associative algebraic structure. Yes it annoying in that we need to write down the parenthesis, but that's it. It's an annoyance.
 
  • #5
pwsnafu said:
Hardly. Order of operations is "needed" only because frankly we can't be bothered writing the parenthesis ##(3 \times 4) + 5## every time we have more than two operations in an expression. It's a notational convention, and has nothing to do with the underlying mathematics. The only part that is "crucial" is the parenthesis.

Just look at any non-associative algebraic structure. Yes it annoying in that we need to write down the parenthesis, but that's it. It's an annoyance.

Indeed.
Auxiliary rules are meant to..auxiliate, not have independent insights.

We MIGHT write 3+4*5+5*(2+1) as A(A(3,M(4,5)),M(5,A(2,1)),

where A(a,b) is the addition operator, and M(a,b) the multiplication operator, but it ought to visually intuitive why we do NOT, usually, write A(A(3,M(4,5)),M(5,A(2,1))
 
  • #6
I understand your point, of course, but I don't agree with the conclusion. I do believe I know less math than the two of you, but is there a good reason ##3 \times 4 +5## implies ##(3 \times 4) + 5## rather than ##3 \times (4 + 5)##?

Isn't it that we've decided that with a lack of parenthesis multiplication goes first? Based on that, if that is correct, I would say something like PEMDAS is very much a rule.
 
  • #7
We wouldn't be in this quagmire if the Indians or Arabs didn't invent the decimal fraction. I guess we have to live with it.
 
  • #8
boit said:
We wouldn't be in this quagmire if the Indians or Arabs didn't invent the decimal fraction. I guess we have to live with it.

What quagmire, exactly? I'm not entirely sure what the problem is supposed to be.
 
  • #9
boit said:
We wouldn't be in this quagmire if the Indians or Arabs didn't invent the decimal fraction. I guess we have to live with it.

You are seeing the entire 1=0.999999... as a huge problem, when it really isn't a problem at all. It's just a consequence of the axioms and definitions we put forward. It really doesn't change math or make it more difficult.
 
  • #10
boit said:
We wouldn't be in this quagmire if the Indians or Arabs didn't invent the decimal fraction. I guess we have to live with it.
That's right. We would, instead, still be doing arithmetic with counters. However, this entire "0.999..." vs "1.0" "quagmire" is, at worst, a mudpuddle that most people step over without ever noticing it.
 
  • #11
boit said:
How do people end with .9r?
Apart from trying to derive square root of numbers using Newton's method, the biggest culprit is how we employ the authorised order of operation in mathematics i.e. BODMAS, PENDMAS e.t.c. Specificaly where we place the quantities.
I saw an equation being used to 'reduce' .9r into 1 (or is it to collapse it into 1?) that went thus
x=.9r
10x=9.9r
subtracting the first equation from the second and we get
9x=9
dividing each side by 9 we find x=1 proving .9r is actually (equal to) 1.
Here is the problem. If we decide to start with division before multiplication, we end up in the same hole that we're trying to get ourselves out of.
Check this. (1/9)x9=?
Now let's divide 1 by 9, what do we end up with? .111 . . . .i.e. 0.1r
multiply that by 9 and we end up with .9r
If we could only have re-arranged the equation to be 1x(9/9)
That way we will avoid unnecessary controversies (but again just like in tabloids, maths seems to thrive in controversies).

.999999...=1.

PEMDAS is a convention.

There are controversies in math. These however aren't among them.
 
  • #12
johnqwertyful said:
.999999...=1.

PEMDAS is a convention.

There are controversies in math. These however aren't among them.

To the uninitiated (like me) it is.
 
  • #13
boit said:
To the uninitiated (like me) it is.

No it is not a controversy, to you it is a confusion.
 
  • #14
Yep, if we divide by 9 instead of multiplying by 10, we end up deeper in the same hole.

But why do that if we can easily grab the vine of multiplying by 10, and then we just need to pull to get out of the hole?
 

FAQ: Avoiding division absurdities.

1. What are some common division absurdities?

Some common division absurdities include dividing by zero, dividing a number by itself, and dividing a number by a very large or very small number.

2. Why is it important to avoid division absurdities?

Division absurdities can lead to undefined or infinite results, which can cause errors in calculations and make it difficult to interpret data. It is important to avoid them to ensure accurate and meaningful results.

3. How can division absurdities be avoided?

Division absurdities can be avoided by checking for potential division by zero before performing the calculation, using appropriate data types and precision in calculations, and considering the context and limitations of the data being divided.

4. What are some real-world examples of division absurdities?

One real-world example of a division absurdity is trying to divide a cake into zero pieces, as this is impossible and would result in an undefined outcome. Another example is trying to divide a distance by time when the time is equal to zero, as this would result in an infinite speed.

5. How can understanding division absurdities benefit scientific research?

Understanding division absurdities can help scientists avoid errors in calculations and ensure the accuracy and validity of their research. It can also help them recognize any limitations or inconsistencies in their data, leading to more meaningful and reliable results.

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