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6.2 Approximation

6.12   Definition ( approximates .) Let be a positive number, and let and be arbitrary numbers. I will say that approximates with an error smaller than if and only if

Remark: If approximates with an error smaller than , then approximates with an error smaller than , since .

6.13   Definition (Approximation to decimals.) Let , and let be real numbers. I will say that approximates with decimal accuracy if and only if approximates with an error smaller than ; i.e., if and only if

6.14   Notation. If I write three dots () at the end of a number written in decimal notation, I assume that all of the digits before the three dots are correct. Thus since I have , and with 4 decimal accuracy.

6.15   Example.

and

Hence

and

Hence approximates with an error smaller than , and approximates with 2 decimal accuracy.

6.16   Example. We see that

and

but there is no two digit decimal that approximates with 2 decimal accuracy.

6.17   Example. Since

we see that approximates with 4 decimal accuracy, even though the two numbers have no decimal digits in common. Since

we see that does not approximate with 4 decimal accuracy, even though the two numbers have four decimal digits in common.

6.18   Theorem (Strong approximation theorem.) Let and be real numbers. Suppose that for every positive number , approximates with an error smaller than . Then .

Proof: Suppose that approximates with an error smaller than for every positive number . Then

Hence

i.e., . But , so it follows that , and consequently ; i.e., .

Next: 6.3 Convergence of Sequences Up: 6. Limits of Sequences Previous: 6.1 Absolute Value   Index
Ray Mayer 2007-09-07