Mahesh Godavarti
3
I think what you are trying to understand is why decimal representations of some numbers do not terminate. It might be useful to read https://www.qalaxia.com/#/viewDiscussion?messageId=5a05ab02a8af23ea0657b31f for an answer to a simpler question about why 1/3 = 0.33333 .... We can use a similar analogy to understand why \pi has a non-terminating representation. \pi is special because it occurs with circles. When we are trying to find a decimal representation of \pi what we trying to do is: exactly cover a circle using squares and a series of sub-squares. Here is a thought experiment that you can do. Imagine you have an ice cylinder of radius 1cm and height 1cm on the left side of the beam balance. Your goal is to balance the beam balance using 10 ice cubes of dimensions 1cm X 1cm X 1cm. The mass of the ice cylinder on the left is \pi units (where 1 unit = 100g). Similarly, the mass of each ice cube is 1 unit. Essentially, we are trying to get a combination of cubes and sub-cubes that approximate \pi units of mass. You are also given a slicer that slices anything that you feed it into 10 equal sized cubes. The way you balance is: 1. Keep loading the right side of the balance with cubes till the right side just becomes heavier than the left side 2. Throw away the left over ice cubes that you didn't load and are not needed 3. Unload the last cube you loaded on the right side and feed it into the slicer 3. Go back to step 1 with the 10 new cubes. You will notice the following. STEP 1: The left side will be heavier as you load up to 3 ice cubes on the right side. The 4th one will make the right side heavier. You will throw away the 6 remaining cubes that you didn't load and won't be using going forward. You will unload the 4th one and feed it into the slicer. You will have 3 cubes from STEP 1 on the right side of the beam balance. --------------------------------- STEP 2: The left side will be heavier as you load up to 1 ice cube on the right side. The 2nd one will make the right side heavier. You will throw away the 8 remaining cubes that you didn't load and won't be using going forward. You will unload the 2nd one and feed it into the slicer. You will have 3 cubes from STEP 1 + 1 cube from STEP 2 on the right side of the beam balance. -------------------------------- STEP 3: The left side will be heavier as you load up to 4 ice cubes on the right side. The 5th one will make the right side heavier. You will throw away the 5 remaining cubes that you didn't load and won't be using going forward. You will unload the 5th one and feed it into the slicer. You will have 3 cubes from STEP 1 + 1 cube from STEP 2 + 4 cubes from STEP 3 on the right side of the beam balance. --------------------------------- So on and so forth. This process will never end. You get the idea that we have approximated \pi to the hundredth decimal place as 3. 14 . The question you might have is if I get another slicer that gives me a different number of slices than 10, then will this process end? The answer is an emphatic NO.
Vivekanand Vellanki
0
Long ago, people were trying to estimate the circumference of a circle. They found that for all circles, the ratio of the circumference and diameter was the same - irrespective of how big/small the circle was. They called this ratio \pi. When they actually computed \pi, they found that the decimal did not terminate. They also found that the decimal did not repeat itself, i.e. the individual numbers in the decimal do not repeat. This means that the number cannot be represented as a rational number. Mathematicians use the term irrationals number to refer to decimals that do not terminate, and do not repeat. \pi is a famous irrational number, and is part of special class of irrational numbers called the transcendental numbers. Transcendental numbers are those irrational numbers that can never be roots of polynomials with rational coefficients. E.g. \sqrt{2} is an irrational number, but is not a transcendental number as it is the root of the polynomial x^2 - 2 . References: https://www.scientificamerican.com/article/what-is-pi-and-how-did-it/