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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Imaginary Numbers - History and Commentary [09/04/1997]

Some history and a different approach to imaginary numbers; Hamilton's approach.

The Infamous .999... = 1 [01/12/2002]

Can we give infinity a value?

Infinite Sets [07/17/1997]

How do you prove that there are more rational numbers than negative integers? How can you tell if an infinite set is countable or uncountable?

Infinity in Undefined (Divide by 0) Situations [9/5/1996]

Elasticity is an inverse slope. If a graph of elasticity is a horizontal line, it is referred to as infinitely elastic. But, wouldn't this really be undefined because it is a divide by zero situation?

Introduction to Bases in Math [02/27/2002]

Rewrite the base 10 numeral in base 5: 13. I don't understand.

Is 120 a Perfect Number? [10/15/1997]

Is 120 a perfect number? It seems to fit all of the criteria.

Mod, Modulus, Modular Arithmetic [05/14/2003]

What does 'mod' mean?

Modular Forms and Elliptic Curves: Taniyama-Shimura [10/30/1997]

I watched a PBS show on Fermat's last theorem and they kept talking about modular forms and elliptic curves and how they are related. What are they, and how so they relate to one another?

Names of Bases (Octal, Hexadecimal, etc.) [04/15/2002]

I don't know the names of the bases after nonadecimal.

Natural Numbers, Positive Integers [04/07/1997]

Why is the set of natural (counting) numbers, i.e. {1,2,3,...}, different from the set of positive integers {1,2,3,...}?

Negative Bases [06/10/1999]

Are there such things as negative bases?

Numbers [12/10/1997]

How do integers and whole numbers, rational numbers, transcendental numbers, and counting numbers relate to each other?

One Million Seconds [09/10/2001]

What is a million seconds in weeks, days, hours, minutes, and seconds? What does unit conversion have to do with number bases?

Palindromic Squares [07/28/1997]

Do you know any numbers besides 14641 where both the number and its square root read the same left to right as right to left?

Perfect Numbers [12/07/1996]

What are the first 10 perfect numbers? Is there a formula for getting a perfect number?

Perfect Numbers [10/17/1996]

Is there any way, other than by trial and error, to figure out what the perfect numbers are?

Planes and Lines [10/26/1996]

Do planes and lines contain the same number of points?

Primes and Perfect Numbers [08/08/1997]

Are there infinite numbers of prime and perfect numbers?

Proof by Induction [4/3/1996]

I was given a proof by my math teacher: by mathemetical induction, prove that i(nCi) = n2^n-1.

Properties and Postulates [08/04/1999]

How do you discover or create a property? What is the difference between a property and a postulate? Do we have to prove all properties?

Prove That -(-a) = a [09/11/2001]

How do you prove that -(-a) = a using the properties of real numbers? What about -1 * -1 = 1?

Pythagorean Triples [11/18/1997]

Could you explain how pythagorean triples work, are calculated, etc.?

Real Number Terminology [12/04/1996]

What does it mean to be non-existent over the reals?

Remainder Problem [10/27/2001]

What number less than 500 produces remainder 4 when divided by 5, remainder 7 when divided by 9, and remainder 9 when divided by 11?

Remainders of 1, 2, 3, 4 [10/09/2001]

Find the smallest whole number that when divided by 5, 7, 9, and 11 gives remainders of 1, 2, 3, and 4 respectively.

Representing Numbers in Different Bases [08/05/1998]

How did they get the 2 x 3 cubed? Plus some number less than 3 cubed?...

Reversing a Number by Multiplying by 9 [08/23/99]

When some numbers are multiplied by 9, why is the result the reverse of the original number?

Significant Non-Zero Digits [11/27/2001]

How many significant digits are there in a number with no non-zero digits? Example: 00.000 Are there any?

Size of Infinity [07/22/1997]

How can one infinity be bigger than another infinity?

Smallest Value of N!; Factorial Table [11/07/2001]

If N! ends in exactly 3 zeros, what is the smallest possible value of N?

Why are Operations of Zero so Strange? [03/17/1997]

Why do we say 1/0 is undefined? Can't you call 1/0 infinity and -1/0 negative infinity? Why not? What is 0 * (1/0)? What is the quantity 0^0?

Why a Zero Exponent Equals One, and Changing Number Bases [9/26/1995]

Why is any number to the zero power equal to one? Could I have some information on hexadecimal and binary for my classes?

Why Isn't Infinity a Number? [2/15/1995]

The concept of infinity is not considered a number. Why?

Why is Zero the Limit? [02/25/2002]

Why is zero called the limit of the terms in the sequence the limit of 1 over n, as n approaches infinity, equals zero?

x Factorial and the Gamma Function [05/29/1998]

What is x! when x is 0, negative, or not a whole number?

0 As the Denominator [01/19/2002]

Can zero over zero equal anything?

1000 $1 Bills in 10 Envelopes [12/11/2002]

Put 1000 $1 bills into 10 envelopes in such a way that someone can ask you for any amount of money from $1 to $1000 and you can give it to him through a combination of the envelopes.

1000 Lockers [11/06/1997]

The 1st student opens all 1000 lockers, the 2nd student closes lockers 2,4,6,8,10, etc., the 3rd student opens lockers closed and closes lockers open on lockers 3,6,9,12,15...

1+1 Theorem [8/5/1996]

What is the 1+1 theorem?

1997M [08/12/2001]

Find all composite positive integers M such that the product of 1997 and M has exactly four divisors.

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