Add new notes from 11.10.2024 and earlier

notes/20241012151815.md

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+# What parameters are there for floating point number representations?
+Floating point number representations $\mathbb{F}$ [[20241010165723]] have 5 parameters.
+$\mathbb{F}(b,p,e_{min}, e_{max}, denorm)$.
+
+* b stands for the base of the number
+* p stands for the length of the mantissa
+* $e_{min}$ stands for the minimal exponent
+* $e_{max}$ stands for the maximal exponent
+* denorm stands for "this number is denormalized"
+
+#gds #floatingpoint

notes/20241012152108.md

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+# What are the parameters for the IEEE 754 single and double precision number representations?
+The parameters for IEEE 754 single and double precision floating point number representations [[20241010165723]] are as follows:
+
+|           | single | double |
+|-----------|--------|--------|
+| b         | 2      | 2      |
+| p         | 24     | 53     |
+| $e_{min}$ | -126   | -1022  |
+| $e_{max}$ | +127   | +1023  |
+| denorm    | true   | true   |
+
+#gds #floatingpoint

notes/20241012152603.md

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+# What is the implicit first bit? 
+The implicit first bit refers to the fact, that in floating point number representations [[20241010165723]] the first bit of the mantissa can be left out,
+because it is only 0 if the denorm bit is true (so it's always 1 unless denorm is on in which case it is 0).
+
+#gds #floatingpoint

notes/20241012152741.md

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+# What format is the exponent represented in for floating point number representation? 
+The exponent of floating point number representation [[20241010165723]] is in excess notation/form [[20241010151125]].
+
+#gds #floatingpoint

notes/20241012152927.md

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+# How to convert a number to its floating point representation? 
+To convert a number $x$ to its floating point representation[[20241010165723]] (base x) do the following:
+1. convert the number to the base of your desired representation
+2. Normalize (shift so there is only one non-fractional place)
+3. calculate the exponent (in excess notation in binary)
+4. Set the sign bit and fill the remaining bits of the mantissa with zeroes
+
+This would be what you would get in binary
+| 1 bit | 8 bit                       | 23 bit                                       |
+|-------|-----------------------------|----------------------------------------------|
+| Sign  | exponent in excess notation | Mantissa(remaining space filled with zeroes) |
+
+#gds #floatingpoint

notes/20241012153323.md

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+# What are the different ways to round for floating point numbers? 
+Rounding floating point numbers to the next numbers in their representation [[20241010165723]] is pretty simple
+You calculate the boundary value of $\hat{x}$ which is $\frac{x_{n}+x_{n+1}}{2}$ if the number is larger round it to $x_{n+1}$, if it is smaller round it to $x_n$.
+Now you have two possibilities if the value you want to round $x=\hat{x}$:
+1. Round away from zero (which means rounding up/down, depending on the sign of the number)
+2. Round to even (which means to round to the next number with 0 as the last bit)
+
+#gds #floatingpoint

notes/20241012153856.md

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+# What is the fundamental axiom of number theory? 
+Every natural number [[20240910105916]] $x \in \mathbb{N} \leq 2$ is representable as a product of prime [[20241012154438]] numbers. This product is called its prime factorization.
+It is unambiguous except for its order.
+
+#math #adm #numbertheory

notes/20241012154438.md

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+# When is a number prime? 
+A natural number [[20240910105916]] $x \in \mathbb{N}$ is prime if it is only divisible by 1 and itself.
+
+#math #adm #numbertheory

notes/20241012154554.md

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+# What are the prime numbers $\mathbb{P}$?
+The prime numbers $\mathbb{P}$ is the set of all natural number [[20240910105916]], that are prime [[20241012154438]].
+
+#math #gds #numbertheory

notes/20241012154721.md

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+# What is $\nu_p(a)$ ?
+$\nu_p(a)$ is a function, that returns how often a specific prime number $p \in \mathbb{P}$ is present in the prime factorization [[20241012153856]] of a number 
+$a \in \mathbb{N}$
+
+#math #adm #numbertheory