See also the
Dr. Math FAQ:
0.9999 = 1
0 to 0 power
n to 0 power
0! = 1
dividing by 0
Browse High School Number Theory
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Infinite number of primes?
Testing for primality.
What is 'mod'?
- Hexadecimal System [02/15/1998]
I know the binary system using base two, but I don't understand the
hexadecimal system using base 16.
- History of Properties [12/11/2001]
Who invented the properties? For example, the distributive property and
the associative property.
- House of Cards [12/02/2001]
Is there a rule for working out the number of cards you need to build a
house of cards of any size?
- How are Binary Codes Used? [01/25/2001]
I've figured out how the binary system works and how to 'translate' from
binary to decimal codes, but how are binary codes used?
- How Can a Number Raised to the Zero Power Be One? [01/23/2008]
I know that 3^7 * 3^0 is 3^(7+0) or 3^7. That means that 3^0 must have
a value of 1. But how is that possible?
- How Does Base 4 Work? [11/06/2003]
How does base 4 work?
- How Many Mice, Cats, and Dogs? [01/13/2003]
You must spend $100 to buy 100 pets, choosing at least one of each
pet. The pets and their prices are: mice @ $0.25 each, cats @ $1.00
each, and dogs @ $15.00 each. How many mice, cats, and dogs must you
- How Many Primes Are Known? [01/21/2003]
How many prime numbers are currently known?
- How Many Rectangular Solids in a Cube? [09/13/2001]
Is there any standard way of finding out how many different possible
rectangular solids can fit into an 3^3 cube?
- How Many Triangles? [10/30/2001]
If we join one point on each of the three sides of a triangle to make
another triangle, there are three triangles with vertices pointing up.
How many triangles will have vertices pointing up if there are n points?
- A Hundred-Row Number Pyramid [11/19/1998]
Starting with two(1,2) in the first row of a pyramid and adding one more
as you go down the list, what is the last number on the righthand side in
the 100th row?
- A Hundred Thousand Switches, One Defect: How Few Tests? [04/19/2011]
Seeking to minimize the binary tests necessary to identify a manufacturing defect, a
student imagines representing the problem as a two-dimensional array, and
pursuing that strategy into higher dimensions. Doctor Anthony introduces a bit-wise
approach for determining the least number, then proceeds through a smaller example
to demonstrate the general method for solving such problems of efficiently identifying
the one bad apple.
- Identity Element [10/12/2001]
What is an "identity element"?
- If n^2 is Even, n is Even [02/21/2002]
I have to show that if n^2 is even then n is also even.
- If N is Odd [12/05/1997]
Prove that if n is odd, then 8 divides (n^2-1).
- Imaginary (Complex) Numbers [02/26/1998]
What are imaginary numbers?
- Incommensurable Numbers [07/25/2001]
What is an incommensurable number?
- Inconstructible Regular Polygon [02/22/2002]
I've been trying to find a proof that a regular polygon with n sides is
inconstructible if n is not a Fermat prime number.
- Increasing and Decreasing Subsequences; Pigeonhole Principle [03/21/2000]
How can I prove that there exists an increasing OR decreasing subsequence
of length n+1 or more in any list of (n^2)+1 distinct integers?
- Indeterminate Forms [09/18/1997]
What is infinity divided by infinity?
- Indeterminate Forms [04/23/2001]
Concerning the indeterminate forms such as 0/0 and infinity/infinity, why
is one to the infinite power considered an indeterminate form?
- The Indeterminate Nature of 0/0 [12/21/2000]
I have a theory that 0/0 = any number, and is not "indeterminate" as is
traditionally claimed. Can you explain the flaw in my thinking, and the
"indeterminate" nature of 0/0?
- Indirect Proofs [01/30/1997]
Give a proof that if r is any nonzero rational number, and s is any
irrational number, then r/s is irrational.
- Induction on .999... [10/19/2000]
In the FAQ proof that .999... = 1, how can you multiply .999... by 10 if
you can never get to the furthest right value? Can you show me an
induction proof that this works?
- Induction Problem [10/14/1997]
Use math induction to prove that (1+2+3+...n)^2 = 1^3+2^3+3^3...n^3.
- Induction Proof of Series Sum [02/03/2001]
How can I prove that for all n greater than 2, the sum 1/(n+1) + 1/(n+2)
+ .. + 1/(2n) is greater than or equal to 7/12?
- Induction Proof with Inequalities [07/03/2001]
Prove by induction that (1 + x)^n >= (1 + nx), where n is a non- negative
- Induction With Binomial Coefficients [10/16/2000]
Prove that the sum from i = 1 to n of (i+k-1 choose k) equals (n+k choose
- Inductive Proof of Divisibility [06/25/2002]
How do you prove that for any integer n the number (n^5)-n is
divisible by 30?
- Inequality Proof for Greatest Integer [10/27/2001]
If x is an arbitrary real number, prove that there is exactly one integer
n that satisfies the inequalities n equal or greater than x less than
- Infinite Continued Fraction [05/15/2002]
What can you determine about the value of the infinite continued
- Infinity as a Skolem Function [10/28/2000]
Is infinity an absolute concept, a relative concept, or both?
- Infinity Hotel Paradox [09/15/1999]
How can a hotel with an infinite number of rooms, all already occupied,
accommodate the passengers of an infinite number of buses without
doubling them up?
- Infinity Solution [04/12/2001]
Can infinity be the solution to the equation 1 + 2x = 3 + 2x?
- Infinity to the Zero Power [04/28/2001]
Does (infinity)^0 equal 1? Why or why not?
- Integer Iteration Function [12/24/2003]
Let X be a positive integer, A be the number of even digits in that
integer, B be the number of odd digits and C be the number of total
digits. We create the new integer ABC and then we apply that process
repeatedly. We will eventually get the number 123! How can we prove
- Integer Logic Puzzle [04/22/2001]
Two integers, m and n, each between 2 and 100 inclusive, have been
chosen. The product is given to mathematician X and the sum to
mathematician Y... find the integers.
- Integer Root Checking [02/18/2003]
Is there a quick way to check whether a number has any roots that are
- Integers and Complex Numbers [02/27/1997]
Do hyper-reals and octonions exist outside complex numbers?
- Integers and Fractions [03/23/2002]
Give an example of a positive integers p,a,b where p/ab and not p/a and
not p/b. Let m, n, and c be integers. Show that if c/m then c/mn.