The quantity 1/3 is only theoretical

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:18n=6n=(18/3)n=(1/3)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.4=xEach 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.f(x)=x-x(1/3)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)^21+b^2=2b^2=1b=sq. rt. 1b=1I 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.