Fixed Point Numbers

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The Binary Point

Fixed Point numbers are a straightforward and simple approach to express Fractional numbers, utilizing a Fixed Number of bits.

The expression "Fixed Point" alludes to the position of the Binary Point. The Binary Point is similar to the Decimal Point of a base-ten number, however since this is Binary as opposed to Decimal, an alternate term is utilized. In Binary, bits can be either 0 or 1 and there is no different image to assign where the Binary Point lies. Be that as it may, we envision, or accept, that the Binary Point lives at a Fixed area between assigned bits in the number.

For example, in a 32-bit number, we can expect that the Binary Point exists specifically between bits 15 (15 in light of the fact that the primary piece is numbered 0, not 1) and 16, giving 16 bits for the Whole number part and 16 bits for the Fractional part. Note that the most significant bit in the whole number field is generally designated as the sign bit leaving 15 bits for the whole number's magnitude.

Place Value Method

Not all numbers can be represented exactly by a fixed-point number, and so the closest approximation is used.

Fixed point.png


In the above example the number stored would be : 32 + 8 + 2 + 1 + 0.25 + 0.03125 + 0.0078125 + 0.00390625 = 43.2869375


Repeated Multiplication

You can convert any decimal fraction to a binary fraction. Multiply the fractional part of the number by 2. Take the integer part of the result (1 or 0) as the first bit. Repeat this process with the result until you run out of patience. For example, to convert 0.3568 into fixed point binary with 8 bits to the right of the binary point,


Fixed point.gif


0.3568 is .01011011

Revision

1. What is the correct binary for the number 15.375?

1111.0110
1111 = 15 0110 = 0.25 0110 = 0.0625 0.25 + 0.0625 = 0.3125
1101.1010
1101 = 13 1010 = 0.5 1010 = 0.125 0.5 + 0.125 = 0.625
1001.1001
1001 = 9 1001 = 0.5 1001 = 0.0625 0.25 + 0.0625 = 0.5625
0111.1011
0111 = 7 1011 = 0.5 1011 = 0.125 1011 = 0.0625 0.5 + 0.125 + 0.0625 = 0.6875

2. What is the correct binary for the number 13.1875?

1111.0101
1111 = 15 .0101 = 0.3125
1011.1111
1011 = 11 .1111 = 0.9375
1010.0111
1010 = 10 .0111 = 0.4375
1101.0011
1101 = 13 .0011 = 0.1875

3. Convert 7.78125 into binary.

111.01100
111 = 7 .01100 = 0.375
111.11001
111 = 7 .11001 = 0.78125
100.01001
100 = 4 .01001 = 0.28125
101.01101
101 = 5 .01101 = 0.40625

4. convert 1111.1010 into denary

15.625
15 = 1111 .625 = 1010
11.375
11 = 1011 .375 = 0110
13.725
13 = 1101 .725 = 1011
10.855
10 = 1010 .855 = 1101

5. What is the correct binary for the number 10.5

0110.0001
0101.1101
1010.1000
1110.1001

6. What is 0101.0101 in denary ?

5.3125
12.9678
7.8665
3.3455
Denary is the standard number system

7. Is 0011.1000, 3.5 in Denary ?

Yes
No

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