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introducing 0th power (Topic)

Let $ a$ be a number not equal to zero. Then for all $ n \in \mathbb{N}$, we have that $ a^n$ is the product of $ a$ with itself $ n$ times. Using the fact that the integer 1 is a multiplicative identity, ( $ a\cdot 1=a$ for any $ a$), we can write

$\displaystyle a^n \cdot 1=a^n=a^{n+0}=a^n\cdot a^0, $
where we have used the properties of exponents under multiplication. Now, after canceling a factor of $ a^n$ from both sides of the above equation, we derive that $ a^0=1$ for any non-zero number.



"introducing 0th power" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: empty product
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Cross-references: equation, sides, factor, multiplication, exponents, properties, multiplicative identity, integer, product, number
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This is version 5 of introducing 0th power, born on 2003-01-31, modified 2004-11-08.
Object id is 3948, canonical name is Introducing0thPower.
Accessed 4687 times total.

Classification:
AMS MSC: 00A05 (General :: General and miscellaneous specific topics :: General mathematics)
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