Carl's Hacking Notes

User Tools

Site Tools

Trace:
numeric

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Next revision		Previous revision
numeric [2025/02/27 17:33] – created carlnumeric [2026/03/14 21:01] (current) – carl
Line 1: Line 1:
 ====== Numeric Types ====== ====== Numeric Types ======
  
 +Numeric types represent [[https://en.wikipedia.org/wiki/Real_number|Real]] numbers in a computer.  Because computers have limited space, most numeric values are approximations of a real value.  
  
 Math in a computer is a combination of a storage of data, and a collection of operators.   Interpretation of the bits is up to the operators.  When we talk about a type of a number, we are usually referring to the operations we can do on them, rather than the storage. Math in a computer is a combination of a storage of data, and a collection of operators.   Interpretation of the bits is up to the operators.  When we talk about a type of a number, we are usually referring to the operations we can do on them, rather than the storage.
Line 7: Line 8:
  
   * Signed Integer (32 bits) - These usually have fast support in the processor.  Sometimes called an int.   * Signed Integer (32 bits) - These usually have fast support in the processor.  Sometimes called an int.
-    * Two's Complement - Represents Numbers as a string of bits.  Bit Vector addition results in (xor(N) + 1) + N = 0 +    * Two's Complement - Represents Numbers as a string of bits.  Bit Vector addition results in (inv(N) + 1) + N = 0 
-    * One's Complement - Represents Numbers as a string of bits.  Bit Vector addition results in (xor(N)) + N = 0+    * One's Complement - Represents Numbers as a string of bits.  Bit Vector addition results in (inv(N)) + N = 0
     * Signed Magnitude - Represents Numbers as one bit for the sign, and the remaining as an unsigned magnitude. N = (-1)^S x M     * Signed Magnitude - Represents Numbers as one bit for the sign, and the remaining as an unsigned magnitude. N = (-1)^S x M
   * Signed Integer (8 bits) - Sometimes called a byte.  Sometimes called an octet in the context of networking.   * Signed Integer (8 bits) - Sometimes called a byte.  Sometimes called an octet in the context of networking.
Line 33: Line 34:
  
  
 +As a consequence of their representation, most operations on numeric types are not true to their eponymous  mathematical function.  For example, adding two Int32 numbers can overflow, resulting in a wrong answer.  Also, most floating point operations are not associative: (a + b) + c != a + (b + c).  This is usually uncommon enough that it isn't a problem, but care should be taken to programming defensively.  
  
 +
 +
 +===== Support in Languages =====
 +
 +==== Java ====
 +Main: [[java:numeric|Java Numeric]].
 +
 +
 +
 +===== Numeric Observations =====
 +
 +When dividing two quantities, the quotient and the remainder have different units.   For example, 10 apples divided by 3 apples is 3, but the remainder is 1 apple.  Note that the quotient unit changed.
 +
 +The units may be different too.  Consider calculating the quotient and remainder of 10 meters in 3 seconds.
 +
 +A = 10 meters
 +
 +B = 3 seconds
 +
 +Q = floor(A / B)
 +
 +R = A % B
 +
 +This maintains the identity A = Q * B + R.  Thus:
 +
 +Q = 3 meters/second
 +
 +R = 1 meter
 +
 +
 +
 +
 +===== Taylor Series =====
 +The Taylor Series for the exponential function diverges quickly if less than 100 terms are used.  These terms tend to have enormous numerators and denominators.   The numerator is usually x^100, and the denominator is 100!.   Floating point math is not suited to represent these values as the number of terms exceeds 100ish.
  
numeric.1740677628.txt.gz · Last modified: by carl