The quantity 1/3 is only theoretical
Debate Rounds (3)
I will only be rebutting so post your arguments in the first round for your reasoning. Good luck.
A Real Number is a number that can be found on the number line. This includes rational and irrational numbers. The quantity (1/3) is real because it can be found on the number line. The quantity (1/3) is not an imaginary or complex number because it is not being multiplied by the imaginary unit i. A "not real number" and a "theoretical" number are two different concepts.
Theoretical is concerned with or involving the theory of a subject area rather than its practical application. I already proved (1/3) is real, and I will now prove (1/3) is pracitcal.
Consider the equation 3x=1. This can be read as "3 times a number equals 1". Let's say I wanted to know what this number is. I would divide both sides of the equation by 3. It would look like this:
I derived the quantity (1/3) because I divided both sides of the equation by 3 to get x by itself. 1 divided by 3 is the quantity (1/3). If I plug (1/3) back into my equation this happens:
(3*1)/3=1; The 3 in the numerator cancles out the the three in the denominator leaving:
1=1, which is a true statement.
(1/3) is the solution to this equation. It is a number I can substitute in for x in the equation above to satisfy the equation. This makes it practical because I can use the number (1/3) to solve an equation. Even a number like the square root of 2 is practical because it can be used to solve an equation such as x^2-5=-3. The square root of 2 is a solution. I'll show that below:
(sq. rt. 2)^2-5=-3
(also negative square root of 2 is a solution to this equation because neg. sq. rt. 2 squared is also 2).
(1/3) is rational because it can be written as the quotient or fraction of two integers p and q with q not equalling zero.
To further my argument I will give a real life situation of this. Let's say you and 2 friends order a pizza. How much of the pizza would each of you recieve? Let p be people. Because there's only 1 pizza and 3 people eating the pizza, we can write and solve an equation for p.
Each of you would recieve (1/3) of the pizza. If each of you put your fair share back together, then if would form 1 whole pizza. This is a practical application of (1/3).
Thank you to con for accepting!
I should have explained this clearer though. I am not arguing by the definition per say of what mathematicians would argue by definition to be "real", just the logic of what is applicable quantitatively. So for the rest of the debate we can use the term theoretical if you like. In this debate I am also going by the base 10 decimal system to avoid confusion in case you were going to be arguing in another base number.
On an actual base 10 number line defined as 1 unit, let's say consisting of no matter to avoid confusion, only an approximation of 1/3 can exist. The reasoning? Well, look at it this way.
Here is why. If there was a value that actually went into one three times, then it would be the same as say 18/3=6
The problem is that in 10 base decimal system, you will never sucessfully divide 1 into three equal parts.
By saying 3x=1
You are using the theoretical 1/3 because by saying
You assume there are infact three equal parts of 1 that will in fact add up to equal 1.
But quantitively, 1/3 gives you an aproximation only. While it is infinitely close however, it will not reach 1/3, because 3s do not add up to ones
This is silly but just to help explain
The pattern goes on infinitely, and will always infinitely draw closer, but will never reach 1.
Allbeit, since the number is infinitely close to a third of one, we use it as such, but in theory.
Saying there is a 1/3 of 1 is theoretical becuase it simply does not exist, and because the version we use in mathmatics is infinitely close to the correct value with an infinitely small percent error we round it to the term 1/3 of 1 since it effects calculations to infintiely small margin.
But you have to understand, I'm only arguing there is no 1/3 of any base 10 numbers, ie 10, 100, 1000 etc
An easy example of this is that 1/3 of say 90 is infact 30. Because 90 is divisible evenly by 30, you get an exact answer. 30 times three does equal 90. You can plug that back in as 30(3)=90 and your answer is not theoretical becuase there is an exact 1/3 of 90 unlike one where you are forced to round for terms of practicality.
The 18/3 example I used does work out quantitatively becuase:
does equal 6 becuase the zeros will infinitely cancel because 0+0+0=zero so the number ends but .33333s will never end, and will continue to grow more complex for eternity
But back to the base 10 number line defined as 1 unit, you will not be able to pinpoint 1/3, because doing so would indicate you have found a point where there is an exact third of one, where one divided by 3 gives you a rational answer, which we have already disproved. I understand you may be arguing from a traditional numberline they show you in school, but you must understand on that line is just an approximation of where it would be in theory.
Back over to con
If we are using a base 10 number line then I could see why you would say the quantity (1/3) is theoretical. I agree that you cannot divide 1 into three equal parts. However, we can divide other quantities into three equal parts. You used 18/3=6. It is true that 18 can be divided into 3 equal parts: 6,6, and 6, but let's say we didn't know that. We could use algebra to figure that out. Let n be a variable. The equation looks like this:
So really, the quantity (1/3) is really just another way of saying "divided by three". Multiplying by the reciprocal of a number is the same as dividing by the number.
Now yes, (1/3)=.333333333 repeating. It is not tangible to have exactly (1/3) of 1 thing in the world, but that doesn't mean you can't have (1/3) of anything.
Let's say you have a 12 inch piece of string that you wish to cut into 3 equal parts. How long would each piece be? Well we can write an equation 12=3x, where x is the length of each piece of string. In order to solve you could either divide both sides of the equation by 3, or multiply both sides by (1/3).
12(1/3)=3x(1/3); on the right side of the equation the 3's would cancel out leaving just x.
Each piece of string would be 4 inches long. Notice 4 is the same as 12 divided by 3, or multiplied by (1/3). That's how we got from 12 to 4. 4 is exactly (1/3) of 12. If you reattached the 3 pieces of 4-inch string, then you would once again have a 12 inch piece of string. If I asked for (1/3) of your piece of string, you could give that to me. One-third of 12 is tangible in the physical world. By you saying the quantity (1/3) is only theoretical means what just happened here is outrageous and not allowed.
I'll give you another example of this. I can't give you one-third of a dollar, but that doesn't mean it doesn't exist. Say you earned a paycheck that varied in amount and you had to give (1/3) of it to your ex- wife for child support (can't really think of a good example). Well, what if you earned exactly $1000 in your paycheck. You can't give exactly (1/3) of that $1000 to your wife, but (1/3) still applies. I described to you a function in which (1/3) is an argument. f(x) is the amount of money you retain after child support, and x is the amount of money in your paycheck.
When you got a paycheck of say $1200, then you could give exactly (1/3) for your child support. Your wife would get $400, and you'd be left with $800. Just because a number is not tangible in the real world doesn't mean it isn't practical.
Lastly, I'd like to address this problem geometrically. Let's consider some right triangle with the length of one leg equalling 1 and the hypotenuse equalling the square root of 2. Even though the square root of 2 is irrational, it still is practical. I can use it in an equation, manipulate it, and use it to solve for other information. Observe:
Leg^2+Leg^2=Hyp.^2 (Pythagorean Theorem)
(1)^2+b^2=(sq. rt. 2)^2
b=sq. rt. 1
I used the square root of 2 to solve an equation. I applied the property if a number under a square root gets squared, the squared and square root cancel. Even though sq. rt. of 2 is irrational, I can still apply it and make use of it. Going back to (1/3), even though it has no ending, it is still practical. You can use (1/3) in equations to manipulate and solve, making it practical.
Back to pro.
JuliusMach10 forfeited this round.
My opponent has forfeited round 3. He/She has failed to respond to my argument in the allotted time. Therefore I have won this debate. The quantity (1/3) is practical.
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