Leanne Poon, a 14 year old female from the Internet asks on November 15, 1998,

What does zero divided by zero equal? -undefined because nothing can divide by zero? -or 1 because the same thing divided by the same thing equals 1? -or 0 because zero divided by anything is zero?

viewed 9178 times

A site called Dr. Math that had this answer. The short answer is "undefined", your first guess above. Here's the long answer: The set R of real numbers does not include an object called infinity, although when mathematicians work with the sets of cardinal or ordinal numbers there are objects that correspond to infinity in one of several senses.

The expressions 1/0 and 0/0 are not defined. That doesn't mean they are infinite. The standard definition for the set Q of all rational numbers is that Q is the set of all real numbers p/q, where p and q are integers and q is not zero. That leaves 1/0 and 0/0 not included in Q since they don't fit the definition.

Some people say that 1/0 is infinity as a kind of short hand for what happens to 1/x as x approaches 0. Note that if x approaches 0 from the right, 1/x becomes larger and larger; often we say that 1/x approaches infinity. Note also that 1/x becomes smaller and smaller as x approaches 0 from the left, so that 1/x approaches negative infinity. 0/0 is often used as shorthand for an indeterminate form in which numerator and denominator approach 0. The ratio is not determined in the sense that it can approach almost anything.

sin(x)/x is a 0/0 indeterminate form; as x approaches 0, both x and sin(x) approach 0; it is known that sin(x)/x approaches 1 as x approaches 0. Try calculating sin(x)/x for x = 0.01, 0.001, 0.0001, etc, in radians.

sqrt(|x|)/x is also a 0/0 indeterminate form; as x approaches 0, both sqrt(|x|) and x approach 0; it is known that the ratio becomes unbounded as x approaches 0. It is misleading to say that it approaches infinity since depending on whether x approaches 0 from the right or left, the ratio becomes very large or very small.

From The Math Forum.

Note: All submissions are moderated prior to posting.

If you found this answer useful, please consider making a small donation to science.ca.

- Canadian Nuclear Safety Commission Educational Resources
- National Inventors Hall of Fame
- JUMP Math
- Society for Canadian Women in Science and Technology
- SciQuest e-Solutions for Science
- Natural Sciences and Engineering Research Council of Canada
- Manning Awards for Innovation
- Royal Society of Canada
- Geological Survey of Canada
- Canadian Networks of Centres of Excellence
- Canadian Landscapes at Natural Resources Canada
- Canadian Association of Physicists
- A Century of Innovation
- Understanding Science
- AlphaGalileo
- National Film Board of Canada Youth Science
- PICS Climate Insights 101
- Canadian Association for Girls in Science
- Virtual Library for the History of Science
- The Chemical Institute of Canada
- Canadian Biotechnologist 2.0
- ISI Highly Cited Scientists
- Deep River Science Academy
- Journal of the History of Canadian Science
- Wikipedia
- Innovation Canada
- Mars Society
- Nobel Prize Archive
- science.gc.ca
- Online Science & Engineering Encyclopedia
- CurioCity
- Canadian Nuclear FAQ
- Association of Science Communicators
- Astrofiles
- Wilderness Astronomy