Ask Dr. Math

High School Archive

Browse High School Number Theory
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Diophantine equations.
    Infinite number of primes?
    Testing for primality.
    What is 'mod'?

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Why Are 1^infinity, infinity^0, and 0^0 Indeterminate Forms? [05/08/1998]

Using limits to prove that 1^infinity, infinity^0, and 0^0 are indeterminate forms.

Why Aren't There Negative Prime Numbers? [12/10/1999]

Why can't negative numbers be prime numbers?

Why Can't 0 Divided By 0 Be 0? [11/25/2003]

Why can't you divide 0 by 0? I've thought about it and it seems that dividing zero objects into zero groups will result in zero groups. Why doesn't this work?

Why Casting Out Nines Works [10/13/2008]

Can you explain in simple terms why the Casting Out Nines method works?

Why Does 0^0 = 1 and Not Undefined? [11/30/2007]

Your proof for why x^0 = 1 uses a law which breaks down at x = 0. Then in your definition for 0^0 you side significantly in favor of 0^0 = 1 based on your rule for x^0 = 1 (which was based on a law that breaks down at 0). Based on what I've read I would side in favor of undefined. Are there any more conclusive reasons for siding with 0^0 = 1?

Why Does 0! = 1 ? [12/8/1995]

Why does 0! = 1 ?

Why is 0! 1? [09/14/1997]

Why is zero factorial 1?

Why Use Q and Z? [09/12/2001]

Why is the letter Q used for rational numbers and Z for integers?

Wilson's Theorem [03/03/2002]

I'm looking for a proof for Wilson's theorem: n divides (n-1)! + 1 if and only if n is a prime number.

Would Aliens Use Base 10? [05/13/2002]

If aliens had 6 fingers, would they use base 12?

Write a Sum That Totals 100 Using Digits 1 - 9 [05/30/2004]

Is it possible to arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 so that when added they total 100? Only adding is permitted and the numbers can be rearranged. Each number can only be used once.

Writing Numbers in Bases Greater Than 10 [04/05/2001]

What would 4 x 13 [base 10] look like in base 42? Do all bases above ten use the same method?

Zero and Imaginary Numbers [07/18/2001]

Is it true that zero divided by an imaginary number is zero? How could the answer be in the real number line when the divisor can't be found in the real number line?

Zero and Infinity [04/24/1997]

Why is the quotient of a number divided by zero infinity?

Zero as an Exponent [7/15/1996]

Why does n^0 = 1?

Zero as Denominator [10/22/1997]

Why can't zero be in the denominator for rational numbers?

Zero Laws and L'Hopital's Rule [03/04/1998]

Is zero divided by zero: a) zero, b) undefined, or c) one?

The Zero Power of Two [12/10/1998]

Why is 2 to the 0 power equal to 1? I don't understand how a number can be multiplied by itself zero times.

Zeros between 1 and 222 Million [11/17/2001]

How many zeros will I use if I write down all the numbers from 1 to 222 million? And how can I generalize this?

Zero to a Negative Exponent [05/06/2001]

Is 0^(-3) equal to 0, or is it undefined? We can't determine whether to use the 0^x = 0 rule, or to interpret it as 1/(0^3).

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