math121a-f23:september_6_wednesday
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math121a-f23:september_6_wednesday [2023/09/07 08:02] pzhou |
math121a-f23:september_6_wednesday [2026/02/21 14:41] (current) |
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| can you define multiplication on it? $ ( 1+ x + x^2) (2 + x) = ?$ | can you define multiplication on it? $ ( 1+ x + x^2) (2 + x) = ?$ | ||
| - | Does every nonzero element has a (multiplicative) inverse? For example, $x$ has inverse, $1/x = x^2$. $x+1$ has inverse, we have | + | Does every nonzero element has a (multiplicative) inverse? For example, $x$ has inverse, |
| + | $$ 1/x = x^2. $$ | ||
| + | $x+1$ has inverse, we have | ||
| $$ \frac{1}{1+x} = \frac{1-x+x^2}{(1+x)(1-x+x^2)} = \frac{1-x+x^2}{1+x^3} = \frac{1-x+x^2}{2}. $$ | $$ \frac{1}{1+x} = \frac{1-x+x^2}{(1+x)(1-x+x^2)} = \frac{1-x+x^2}{1+x^3} = \frac{1-x+x^2}{2}. $$ | ||
| + | Does $x-1$ has inverse? | ||
| ------- | ------- | ||
| you may ask: why we care about other ' | you may ask: why we care about other ' | ||
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| + | ==== Exercise ==== | ||
| + | (part of homework) | ||
| + | Read Boas Ch2, section 1 - 9, find 5 interesting problems there and do it. (copy down the problem, so the grader / reader know which one you are doing). | ||
| + | |||
math121a-f23/september_6_wednesday.1694073779.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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