A little clarification on absolute values

In summary, absolute values are positive values that represent the distance of a number from zero on a number line. To calculate the absolute value of a number, you can remove the negative sign if the number is negative. Absolute values are important in mathematics because they allow for easy comparison of numbers regardless of their signs. They can also be decimals or fractions. The difference between absolute value and magnitude is that absolute value is for numbers, while magnitude is for vectors or quantities.
  • #1
relinquished™
79
0
Hello again. I have a (stupid, but I'm not real sure about the answer-type) question. I'm trying to prove that the second order ODE of the simple pendulum y''=-(g/l)sin y is Lipschitz (using norm 1). After doing some evaluating, I came up with

[itex]
|u'-v'| + |\frac{g}{l}||\sin u - \sin v|
[/itex]

All I'm asking is if this is true for all values of u and v:

[itex]
|\sin u - \sin v| \leq |u - v|
[/itex]

All clarifications are appreciated.

Thanks,

Reli~
 
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  • #2
I think you can show that |sin(u)-sin(v)| <= |u-v| using the MVT.
 

What are absolute values?

Absolute values represent the distance of a number from zero on a number line. It is always a positive value.

How do I calculate the absolute value of a number?

To calculate the absolute value of a number, you can remove the negative sign if the number is negative. If the number is already positive, then the absolute value is the same as the original number.

Why are absolute values important in mathematics?

Absolute values are important because they allow us to compare numbers without considering their signs. They are also used in various mathematical operations and equations.

Can absolute values be decimals or fractions?

Yes, absolute values can be decimals or fractions. The absolute value of a decimal or fraction will always be a positive number.

What is the difference between absolute value and magnitude?

Absolute value and magnitude are often used interchangeably, but there is a subtle difference. Absolute value is used for numbers, while magnitude refers to the size or extent of a vector or quantity.

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