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