60 lines
2.4 KiB
Markdown
60 lines
2.4 KiB
Markdown
## Convert numbers to binary
|
|
Number in decimal: 53.7
|
|
### Decimal conversion
|
|
|
|
| Calculation | Remainder/Binary |
|
|
| ----------- | ---------------- |
|
|
| 53 / 2 = 26 | 1 |
|
|
| 26 / 2 = 13 | 0 |
|
|
| 13 / 2 = 6 | 1 |
|
|
| 6 / 2 = 3 | 0 |
|
|
| 3 / 2 = 1 | 1 |
|
|
| 1 / 2 = 0 | 1 |
|
|
So in binary 110101, as the order is in reverse of the decimals
|
|
### Fraction Conversion
|
|
|
|
| Calculation | Non Decimal Portion/Binary |
|
|
| ------------- | -------------------------- |
|
|
| 0.7 x 2 = 1.4 | 1 |
|
|
| 0.4 x 2 = 0.8 | 0 |
|
|
| 0.8 x 2 = 1.6 | 1 |
|
|
| 0.6 x 2 = 1.2 | 1 |
|
|
| 0.2 x 2 = 0.4 | 0 |
|
|
And so on.. so the fraction would be .10110, with 0110 repeating infinitely
|
|
So the final number would be 110101.10110...
|
|
|
|
**Normalization** is the process is the process of adjusting a number so only 1 non zero digit on the left side of a number, i.e. the number is in scientific notation
|
|
|
|
## Floating point number types
|
|
|
|
| Precision | Sign | Exponent | Mantissa |
|
|
| ----------- | ---- | -------- | -------- |
|
|
| single | 1 | 8 | 23 |
|
|
| double | 1 | 11 | 52 |
|
|
| long double | 1 | 15 | 64 |
|
|
Truncation types
|
|
**Chopping**: Omit the numbers that we don't want, looking to the first bit that we want to erase
|
|
**Rounding**: We should take care about the first digit that we want to omit and adjust the 52nd bit
|
|
|
|
**Machine Epsilon**: Difference between the smallest floating point number greater than 1 and 1, i.e. the smallest number that when added to 1, will be different than 1
|
|
|
|
IEEE rounding to nearest role:
|
|
1. If we have zero in the 53rd bit, we will round down
|
|
2. If we have one as the 53rd bit, we will round up
|
|
1. If the 52nd bit is one we will round up
|
|
2. If the 52nd bit is zero will round down
|
|
|
|
Convert a real number to a floating point number:
|
|
1. Decimal to binary number
|
|
2. Justify step: Shift the radix to the right of the left most one, compensate with the exponent. 100.1 -> 1.0001 x_2^3
|
|
3. We do it with respect to p.A d.p 52 numbers ??? I think she means IEEE rounding
|
|
|
|
There is no need to represent the first bit of the mantissa, since it is always 1 with certain exceptions
|
|
|
|
Overflow - exponent greater than 1023
|
|
Underflow - exponent less than 2^-1074
|
|
Normally set to zero
|
|
|
|
Mean value theorem - *Info and graph in slides*
|
|
|
|
Taylor's Theorem |