Unsolved Challenge: Floor function

In summary, the floor function, denoted as ⌊x⌋, is a mathematical function that rounds down a real number to the nearest integer. It is different from rounding as it always rounds down, while rounding can round up or down. Some common applications include converting decimal values to integers, data analysis, and solving mathematical problems. The function can also return negative values. Another similar function is the ceiling function, which rounds up to the nearest integer, and there is also a rounding function that rounds to the nearest integer.
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anemone
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Solve $\{ \sin \lfloor x \rfloor \}+\{ \cos \lfloor x \rfloor \}=\{ \tan \lfloor x \rfloor \}$ for real solution(s).
 
  • #1
x in radian or degree ?
 
  • #1
Hi Kali, $x$ is in radian.
 
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  • #2
anemone said:
Solve $\{ \sin \lfloor x \rfloor \}+\{ \cos \lfloor x \rfloor \}=\{ \tan \lfloor x \rfloor \}$ for real solution(s).
Sorry, but I'm a bit confused. I know what the floor function does but what does the {.} do?

-Dan
 
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  • #2
Sorry Dan for not being clear in my question.(Blush)

{} means the fractional part of $x$, and defined by the formula $\{ x \}=x-\lfloor x \rfloor$.

Hope this clears it up!
 

1. What is the floor function and how does it work?

The floor function, also known as the greatest integer function, takes a real number as input and returns the largest integer less than or equal to that number. For example, the floor of 4.8 is 4, and the floor of -2.3 is -3.

2. What is the difference between the floor function and the ceiling function?

The ceiling function is the opposite of the floor function, as it returns the smallest integer greater than or equal to a given number. For example, the ceiling of 4.8 is 5, and the ceiling of -2.3 is -2.

3. How is the floor function used in mathematics?

The floor function is commonly used in a variety of mathematical applications, such as in number theory, calculus, and computer science. It is often used to round down a number to the nearest integer, and is also useful in solving equations and inequalities.

4. Are there any special properties of the floor function?

Yes, the floor function has several important properties. It is a piecewise constant function, meaning it remains constant on intervals of consecutive integers. It is also a discontinuous function, as it has jumps at every integer. Additionally, the floor function is always equal to or less than the input number.

5. Can the floor function be extended to complex numbers?

Yes, the floor function can be extended to complex numbers by using the real and imaginary parts separately. For example, the floor of a complex number with real part 3.2 and imaginary part 4.7 would be the complex number with real part 3 and imaginary part 4.

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