The scale is accurate to 1 gram. Did I trick you? Can you tell the difference? Suppose your error tolerance is 0. Then if you ask for 1, and I give you 0.
The curve is the number of digits we expand 0. The idea is to expand 0. At some point, no matter what you pick for e , 0. As an aside, 0. The curve represents the idea that we can approximate 0. With limits, if the difference between two things is smaller than any margin we can dream of, they must be the same. This first conclusion may not sit well with you — you might feel tricked.
We seem to be ignoring something important when we say that 0. Things are either grains, or not there.
I interpret 0. The happy compromise is this: in a more accurate dimension , 0. But, when we, with our finite accuracy, try to describe the difference, we cannot: 0.
The question is about exploration. How do we handle them? There are life lessons here: can we extend our mental model of the world? Negatives gave us the conception that every number can have an opposite. And you know what? Expanding our perspective with strange new ideas helps disparate subjects click. When writing, I like to envision a super-pedant, concerned more with satisfying and demonstrating his rigor than educating the reader. This mythical? The distance between two real numbers is just the difference of the larger number minus the smaller number.
So, the distance between the first number in our sequence and 1 is 1 - 0. The distance between 1 and our second number is 1 - 0. The distance between 1 and our third number is 1 - 0. Each term of our sequence gets closer and closer to 1, and for any tiny little distance we want, we can find a number in our sequence — some finite pile of 9's after a decimal point — that is closer to 1 than that tiny little distance. Since we saw above that the numbers in our sequence are always getting closer to 1, this further means that all the subsequent numbers in the sequence will also be closer to 1 than our tiny distance.
To see this in action, here are the first five numbers in our sequence, all getting really close to When an infinite sequence of numbers has this property of getting arbitrarily close to some number — when, for any tiny little distance we choose, we can find some point in the sequence so that every number in the sequence after that point is closer to the target number than that tiny little chosen distance — we say that the sequence converges to that number, called the limit of the sequence.
In the case of an infinite decimal, again standing in for the kind of infinite sequence of terminating decimals we saw above, we identify the sequence with its limit. This is what we mean when we say that 0.
The same idea works for any rational number with a repeating infinite decimal expansion. Something similar happens with irrational numbers that have non-repeating decimal expansions. In general, any infinitely long decimal expansion is thought of as the limit of the sequence of terminating decimals that make up the infinite expansion. So, the reason why 0. Reasonably then, 0.
But 3 0. Then 0. Argument from arithmetic: When you subtract a number from itself, the result is zero. So what is the result when you subtract 0. For shorter subtractions, you get:. Then what about 1. You'll get an infinite string of zeroes. The zeroes go on forever. And 0. Argument from philosophy: If two numbers are different, then you can fit another number between them, such as their average. But what number could you possibly fit between 0. Argument from algebra: The expression 0. Call this numerical value " x ", so 0.
Multiply this equation by ten:. There is no "end" after which to put that zero. Argument from semantics: A common objection is that, while 0.
But what is meant by "gets arbitrarily close"? It's not like the number is moving at all; it is what it is, and it just sits there, looking at you.
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