What are Floating Point Numbers? An Explanation with Examples

[SherlockOhms]

So, I'm not 100% sure if this is actually in the correct forum or whether it should be under Mathematics. Anyway, our lecturer told us that we'd soon cover "Floating Point Numbers". Could someone give me a basic, somewhat simple explanation of what these are? Possibly with a few examples. Just so I have an idea of what to expect. Thanks!

 

[berkeman]

So, I'm not 100% sure if this is actually in the correct forum or whether it should be under Mathematics. Anyway, our lecturer told us that we'd soon cover "Floating Point Numbers". Could someone give me a basic, somewhat simple explanation of what these are? Possibly with a few examples. Just so I have an idea of what to expect. Thanks!

It's a way to represent a wider range of numbers -- basically including exponents in the representation:

http://en.wikipedia.org/wiki/Floating_point

 

[MrTheBard]

If your class is a Computer Science / Programming class, a floating point number could refer to any number with a decimal value. 1 is an integer, 1.0 can be considered a float. This is crucial to understand for computer programming. If it is not for a computer programming class, then the aforementioned post by berkeman provides a good link to wikipedia's definition on floating point numbers.



Gary

 

[SherlockOhms]

Right, I think I've got it. Sort of anyway. Thanks for the help. I guess I'll just have to wait and see what the lecturer is going to show us. Thanks again.

 

[jtbell]

Don't you have a textbook or other recommended reference source?

 

[SherlockOhms]

Nope. Not yet anyway. There's no recommended text for the course and we only began last Monday. Things will become clearer over the next few weeks.

 

[phinds]

Nope. Not yet anyway. There's no recommended text for the course and we only began last Monday. Things will become clearer over the next few weeks.

Surely you DO know what kind of class you are in, no? Is it a computer science class, a physics class, a math class, what?

 

[SherlockOhms]

Of course. Never said I didn't. It's a computer programming class with numerical methods involved.

 

[phinds]

Of course. Never said I didn't. It's a computer programming class with numerical methods involved.

Then read post #3

 

[harborsparrow]

If your class is a Computer Science / Programming class, a floating point number could refer to any number with a decimal value. 1 is an integer, 1.0 can be considered a float. This is crucial to understand for computer programming. If it is not for a computer programming class, then the aforementioned post by berkeman provides a good link to wikipedia's definition on floating point numbers.



Gary

A key thing to remember about floating point numbers, as used in computers, is that they are stored as approximations, unlike integers which can be represented exactly. Thus, one must never test two numbers of type float for equality (as in does "x = y", where both are floating point numbers), because even if they are so closely approximated that both print out as having a value of, say 1.3, the equality test might still fail.

Further, because of how they are stored, it is possible to have issues with overflow and loss of precision when doing calculations with them (depending on the order in which computations are done to the numbers, for example). To learn about these things, read a book on numerical computing (or wait until your professor covers them).

Handling floating point numbers correctly in programming is an important topic, and may take some time to learn.

 

[Mark44]

A key thing to remember about floating point numbers, as used in computers, is that they are stored as approximations, unlike integers which can be represented exactly.

Most floating point numbers are stored as approximations, but some are stored exactly, such as 1.5, 3.75, and -4.875. Any number whose fractional part is made up of a finite sum of powers of 1/2 (within limits based on the size of the mantissa) is stored exactly.

Thus, one must never test two numbers of type float for equality (as in does "x = y", where both are floating point numbers), because even if they are so closely approximated that both print out as having a value of, say 1.3, the equality test might still fail.

Further, because of how they are stored, it is possible to have issues with overflow and loss of precision when doing calculations with them (depending on the order in which computations are done to the numbers, for example). To learn about these things, read a book on numerical computing (or wait until your professor covers them).

Handling floating point numbers correctly in programming is an important topic, and may take some time to learn.

 

[berkeman]

Thus, one must never test two numbers of type float for equality (as in does "x = y", where both are floating point numbers), because even if they are so closely approximated that both print out as having a value of, say 1.3, the equality test might still fail.

And remember in C at least, the check four equality would be "x == y", since == is a comparison and = is an assignment. Too easy to get those mixed up...

 

[SherlockOhms]

Thanks for all the info, guys.

 

[harborsparrow]

I, for one, am not a guy. But I'll take the thank you generically.

 

[SherlockOhms]

'Twas meant generically.

 

[SherlockOhms]

Sorry to dig up this old thread, but could someone recommend a numerical methods book for me to look at? I've got a far better understanding of floating point numbers now, but am getting a little bit stuck with calculating the error between them and a given value, x.

 

[kreil]

Floating point numbers are defined on a grid. In other words, they have finite precision, and if a calculation falls between two floating point numbers, there will be roundoff error. Since the smallest distance between two floats is machine epsilon, the maximum value of the roundoff error (in matlab) is eps/2, or about 1.1102e-16 in double precision. Here is an example in matlab:

>> A = 0.3 + 0.3 + 0.3

A =

    0.9000

>> abs(A - 0.9) == eps/2

ans =

     1

As mentioned in a previous post, this is why you don't compare them directly for equality, but rather to a tolerance.

 

[SherlockOhms]

Floating point numbers are defined on a grid. In other words, they have finite precision, and if a calculation falls between two floating point numbers, there will be roundoff error. Since the smallest distance between two floats is machine epsilon, the maximum value of the roundoff error (in matlab) is eps/2, or about 1.1102e-16 in double precision. Here is an example in matlab:

>> A = 0.3 + 0.3 + 0.3

A =

    0.9000

>> abs(A - 0.9) == eps/2

ans =

     1

As mentioned in a previous post, this is why you don't compare them directly for equality, but rather to a tolerance.

Thanks for that! However, it's not so much the concept of the round off error that I'm struggling with, more so it's calculations associated with it. I've actually posted the specific problem that I've struggled with in the homework and course work section and it'd be great if you could have a look as it'd probably give you a better understanding of what I'm struggling with. It's a worked example to do with finding the floating point representation of 0.2 and then the error between the two values.

 

[SherlockOhms]

Also, could someone let me know if what I did above is ok? I'm not sure making reference to another thread is allowed or not.