How can one divided by zero be infinity because if something remains undivided it will still be there?

Answer

To try to understand the concept of infinity, play with the figures:-

1 divided by 0.1 (or 10 to power of -1)= 10 to power of 1 = 10

1 divided by 0.01 (or 10 to power of -2) = 10 to power of 2 = 100

1 divided by 0.001 (or 10 to power of -3) = 10 to power of 3 = 1000

1 divided by 0.000001 , (or 10 to power of -6) = 10 to power of 6

1 divided by 10 to power of -1000000000) = 10 to power of 9

1 divided by 10 to power of -zillion ) = 10 to power of a zillion

and so on.

This traditional concept of infinity is of piece of string cut in half, then that half cut in half again, and so on, again and again, down to an infinitesimal length which cannot be measured further. It is assumed that since our knowledge and expertise etc is ever-increasing, it will mean that the smallest we can measure gets ever-smaller; 1 divided by an ever-incredibly smaller number will give a correspondingly ever-incredibly large number , and that the bounds of infinity are constantly being expanded. Since there will always be a remainder left over, you are right to be confused.
That all seems very well in theory, but in practice things are now slightly different.
Scientists have now discovered that the piece of string cannot be cut forever into increasingly-small halves, but that there is a minimum length which cannot reduced any further. This length is called the Planck length, and is 1.616252 × 10⁻³⁵ meters. This is the shortest length possible: beyond that there is 'non-locality' . Just as there is an absolute minimum size, so it follows that there must be an absolute maximum size. This means that the universe is actually finite , probably about 10 to the power of 25 meters in diameter.
This information is mind-boggling, and the questioner is perfectly entitled to feel confused about concepts such as infinity which have only been known about by scientists today since the advent of Quantum Mechanics, with the knowledge that Time is a dimension and there are more than the three spatial dimensions.
What is really mind-boggling is how Paul the Apostle knew this in 50-60AD when he wrote of more than three dimensions: count them:-

Paul could not have known this by himself: he had to have been told this fact by God. To paraphrase God :"Folks, you ain't seen nothin' yet":-

For further information, see the Wikianswers question "How fast is Planck time?"
and the Wikipedia article on Planck units.

ANSWER

Your question assumes that if we divide one number by another we are somehow "chopping up" the first number. That view arises because in the lower grades we teach division by using solid matter; for example, how a pie is cut up ("divided") and distributed, or how lines are cut up into sections. But numbers are not solid objects; they are symbols that stand for thoughts, and their numerosity -- their "numberness" -- has no physical existence.

Thus, in simple arithmetic, division is a "thought" problem, not a problem about manipulating solid matter. As a thought problem, division can be understood as successive subtraction of abstract numbers until the result is 0. The problem "6 divided by 2," for example, asks how many times you can subtract 2 from 6 until you reach 0. There is no implication that after this "thought process" the number 6 is somehow "gone." It was never "there" in the first place: it is only a symbol, not a real "thing."

Your question asks how many times we can subtract 0 from 1 -- or from any number-- before the result is 0. As you can see, from this "abstract" viewpoint we could go on subtracting 0 from any number forever without bringing our answer down to 0.

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