Category: Uncategorized
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Exponentiation
is a shorthand for repeated multiplication. For example, . In other words, to compute the value of we multiply three ‘s. . We can generalize: . Since , we can generalize: . . However, . Since is undefined, is also undefined. . Thus is the reciprocal of . Obviously, is undefined. What about ? We…
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Division by zero
For two real numbers and if we can find a unique real number such that then we say and that is the definition of division. A division between real numbers is undefined if we fail to find any or there are more than one choices for . These may happen when . For example, would…
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Sign of product
Sign of a product of two numbers comes down to: , , , and . For example, . Since is the multiplicative identity, by definition: . So, , , and . What should be? By definition: . More generally, . Since is the multiplicative identity, . Assuming negative numbers abide by associative law of multiplication:…
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fractions on a number line
To compare two natural numbers I can count their digits or if they have the same digit-count, I scan the digits from left to right. To compare two fractions I have to think harder. For example, to compare and , I end up comparing and . Every natural number is a multiple of 1, like…
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limit of a function: the beginning
The limit of a function at , , feels a lot like the value of the function . In fact, for simple functions like , , , the two are identical, they evaluate to the same number. Yet, the limit is different from a function’s value in general. With limit, we do not bother looking…
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logarithmic function
Logarithmic function is the inverse of exponential function. Duh! Yet, this all important definition I have stashed too far away from my conscious. By practice it has become second nature to write: , which also helps explain or . When in doubt I might go back to the fact: . But I would have trouble…
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one, two, three, five, four, … : number and counting
Some years ago I bought a book called Algebra by Michael Artin. In the preface there was a warm conversation: “One, two, three, five, four …” “No Daddy, it’s one, two, three, four, five.” “Well if I want to say one, two, three, five, four why can’t I?” “That’s not how it goes.” The conversation…
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ones, tens, hundreds … : place-value number system
Place-value number system like decimals for an example, is better than say Roman numerals. For example, a quick glance is sufficient to declare is bigger than , because has more digits (or symbols). On the other hand, it is not that easy to say which one is bigger: or , actually the latter is bigger…
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continuity: the number line
Continuity is mysterious. Even the continuity of a simple, unbroken straight line is deep mystery. For a starter, consider the interval on a number line. It contains the interval . Yet, there is a one-to-one map between these two intervals, therefore for each point in the larger, containing interval we find a point in the…
unreasonably effective 