Exponent: Difference between revisions
imported>Catherine Woodgold (Adding "See also" section with link to logarithm) |
imported>Catherine Woodgold (→Extension of exponents to fractional and negative values: Moving punctuation to inside math environments) |
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== Extension of exponents to fractional and negative values == | == Extension of exponents to fractional and negative values == | ||
Originally, exponents were natural numbers: it's easy to see the meaning of an expression such as <math>10^3 = 10 \times 10 \times 10</math> | Originally, exponents were natural numbers: it's easy to see the meaning of an expression such as <math>10^3 = 10 \times 10 \times 10.</math> Rules for adding and multiplying exponents were noticed, and to extend the idea to fractions and negative numbers it was assumed that the same rules would apply. To define a meaning for a fractional value such as <math>10^\frac{1}{2},</math> consider that, using a rule for multiplying exponents, | ||
:<math>(10^\frac{1}{2})^2 = 10^{\frac{1}{2}\times 2} = 10^1 = 10</math> | :<math>(10^\frac{1}{2})^2 = 10^{\frac{1}{2}\times 2} = 10^1 = 10</math> | ||
Therefore <math>10^\frac{1}{2}</math> must be <math>\sqrt{10}</math> | Therefore <math>10^\frac{1}{2}</math> must be <math>\sqrt{10}.</math> Values for many other numbers can be worked out similarly using cube roots and so on, and values for all real numbers can then be defined using limits. | ||
To assign meaning to negative values of exponents, note the rule that | To assign meaning to negative values of exponents, note the rule that | ||
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:<math>b^x b^y = b^{x + y}.</math> | :<math>b^x b^y = b^{x + y}.</math> | ||
So, for example, to find the meaning of <math>10^{-3}</math> | So, for example, to find the meaning of <math>10^{-3},</math> consider | ||
:<math>10^{-3}\times10^4 = 10^{-3 + 4} = 10^1 = 10</math> | :<math>10^{-3}\times10^4 = 10^{-3 + 4} = 10^1 = 10</math> | ||
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:<math>10^{-3} = \frac{10}{10^4} = 0.001</math> | :<math>10^{-3} = \frac{10}{10^4} = 0.001</math> | ||
By a similar argument it can be established that <math>b^0 = 1</math> for any base <math>b>1</math> | By a similar argument it can be established that <math>b^0 = 1</math> for any base <math>b>1.</math> | ||
==See also== | ==See also== | ||
Revision as of 09:56, 21 May 2007
An exponent is one of those tiny numbers at the upper right that means you have to multiply something that number of times: for example, means 5 multiplied by itself 3 times or the number 3 is called the exponent.
Extension of exponents to fractional and negative values
Originally, exponents were natural numbers: it's easy to see the meaning of an expression such as Rules for adding and multiplying exponents were noticed, and to extend the idea to fractions and negative numbers it was assumed that the same rules would apply. To define a meaning for a fractional value such as consider that, using a rule for multiplying exponents,
Therefore must be Values for many other numbers can be worked out similarly using cube roots and so on, and values for all real numbers can then be defined using limits.
To assign meaning to negative values of exponents, note the rule that
So, for example, to find the meaning of consider
Therefore,
By a similar argument it can be established that for any base
