Bdtyktl

Suppose I have a fractional decimal number like 0.21

Can the windows calculator convert that to binary?

If I try to type in 0.21 while in programmer mode, it doesn't respond to the decimal point. It's like I can't type in a decimal point in programmer mode.. And I can't type a binary point in programmer mode either when I select binary within programmer mode. The 'point' seems to be disabled.

I am using windows 7. I don't know if this has been enabled in windows 10?

Added

Or how about a simpler example like 2.5 or 5.125 So, a fractional part that can be represented in binary? So for 2.5 that calc should show 010.10 or 1.10*2^3. Or for 5.125 the calc should show 0101.001 or 1.01001*2^5 Can the windows calculator show that?

@Keltari Generally, fraction value CAN be represented in binary, using kind of "binary point" (counterpart of decimal point). In fact, that is how real/double numbers are represented in computers (although usually binary point position is stored, instead of the "point character"). But remember: not all decimal fractions can be represented in binary form exactly - some rounding must be applied then, especially if only limited count of bits can be used (which is always true in real computer science world).

Unfortunatelly, windows calculator in programmer's mode operate on integer values only. A workaround to achieve your goal is like this:

1. In standard mode enter a decimal number (can be fractional) to convert, e.g. 0.21




2. Multiply it by 2 (keys: * 2 =)



3. 
   

   Repeat step 2 (pressing = key alone, again and again), until you receive number which is integer - or very close to integer (see remark below).

   
   
   

   IMPORTANT! Remember, how many times you multiplied initial value by 2 (in step 2 and repeated 3).

   
   



4. Switch to programmer's mode and enter value received after step 3, rounded to integer (unfortunately, that is not done automatically when switching mode, at least in Windows 10's Calculator app).




5. The binary representation of the integer number is shown. Imagine, the binary point is on the most right (after last binary digit). Then, move it left by as many positions as you have done multiplying in step 2 and 3. Add leading 0's as needed.

Example:

• Original value: 0.21
  


• Multiply it by 2 ten times - you get 215,04. Assuming that it is close enough to an integer value (it is arbitrary decision!), you get the rounded integer of 215
  


• Entering 215 in programmer's mode shows its binary representation: 11010111
  


• Moving the binary point 10 places left, gives: .0011010111
  

Remark: As I mentioned earlier, not all decimal fraction values can be represented exactly in binary form, so sometimes you must decide, what precision (approximation) satisfy your needs. In the example above, I assumed that rounding 215,04 to 215 is comparatively small "error" (below 1/5000), so 10 bits was enough to represent the original number. If I needed greater precision, I would make multiplying by 2 not ten, but maybe twenty times (result: 220200,96 rounded to 220201 as an integer) and would get 20-bits binary representation .00110101110000101001 (with "error" below 1/5000000).

BTW Usually count of bits used to represent data in computers is power of 2 - 8, 16, 32, 64, sometimes 48, etc. In such a case, you can simply use faster "multipling by 2" method in the calculator, suggested by you in comment below (*(2^n)).

So the way fractions are represented in memory is according to the IEEE-754 standard for floating point numbers.

A number as simple as decimal 123 when stored as floating point number (e.g. 123.0) is actually stored as 1.921875 * 2^6 = 1.921875 * 64

Think of this as a scientific notation, which is a * 10^b, where a must be [1, 10) (meaning from 1.0 up to, but not including 10, e.g. 9.999...8)

Your number 0.21 is stored in memory as 1.6799999475479126 * 2^(-3) which is equivalent to 0.20999999344348907470703125 (windows calculator shows this as 0.209999993443489075 when calculating manually)

Any calculator that utilises this standard will store numbers in the same format (if it doesn't, excluding different endianness, it probably has a different way of storing floating point numbers, so when passing values from one piece of software/hardware to another, translation must happen, but this is a different topic altogether).

Now, to answer your question, while Windows calculator stores the numbers in this form, it hasn't been designed to show you the intermediate/raw values in the memory. If you really felt like it, you could use a memory viewing tool and spy on values as you type them in the calculator, but again, that is a different topic...

Check out https://www.h-schmidt.net/FloatConverter/IEEE754.html to have an idea of how the floating point numbers are stored and represented.