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--- en:multiasm:cs:chapter_3_11 [2025/01/08 17:49] – [Table] ktokarz
+++ en:multiasm:cs:chapter_3_11 [2025/05/15 06:20] (current) – [Integers] ktokarz
@@ Line 3: / Line 3: @@
 ===== Integers =====
-Integer data types can be 8, 16, 32 or 64 bits long. If the encoded number is unsigned it is stored in binary representation, while the value is signed the representation is two's complement. In two's complement representation, the most significant bit (MSB) represents the sign of the number. Zero means a non-negative number, one represents a negative value. The table {{ref>binarynumbers}} shows the integer data types with their ranges.
+Integer data types can be 8, 16, 32 or 64 bits long. If the encoded number is unsigned, it is stored in binary representation, while if the value is signed, the representation is two's complement. A natural binary number starts with aero if it contains all bits equal to zero. While it contains all bits equal to one, the value can be calculated with the expression {{ :en:multiasm:cs:equation_binary.png?200 |}}, where n is the number of bits in a number.
+In two's complement representation, the most significant bit (MSB) represents the sign of the number. Zero means a non-negative number, one represents a negative value. The table {{ref>binarynumbers}} shows the integer data types with their ranges.
 <table binarynumbers>
 <caption> Integer binary numbers</caption>
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 ===== Floating point =====
-Integer calculations do not always cover all mathematical requirements of the algorithm. To represent real numbers the floating point encoding is used. A floating point is the representation of the value //A// which is composed of two numbers
+Integer calculations do not always cover all mathematical requirements of the algorithm. To represent real numbers the floating point encoding is used. A floating point is the representation of the value //A// which is composed of three fields:
-  * Mantissa (Ma)
+  * Sign bit
-  * Exponent (Ea)
+  * Exponent (E)
+  * Mantissa (M)
 fulfilling the equation
 {{ :en:multiasm:cs:equation_floating.png?200 |}}
-There are two main types of real numbers, called floating point values. Single precision is the number which is encoded in 32 bits. Double precision floating point number is encoded with 64 bits. They are presented in Fig{{ref>realtypes}} while Table{{ref>realnumbers}} shows their properties.
+There are two main types of real numbers, called floating point values. Single precision is the number which is encoded in 32 bits. Double precision floating point number is encoded with 64 bits. They are presented in Fig{{ref>realtypes}}.
 <figure realtypes>
@@ Line 30: / Line 33: @@
 <caption>Illustration of a single and double precision real numnbers</caption>
 </figure>
+The Table{{ref>realnumbers}} shows a number of bits for exponent and mantissa for single and double precision floating point numbers. It also presents the minimal and maximal values which can be stored using these formats (they are absolute values, and can be positive or negative depending on the sign bit).
 <table realnumbers>
 <caption> Floating point numbers</caption>
-^ Precision        ^ Exponent  ^ Mantissa  ^ The smallest                              ^ The largest  ^
+^ Precision        ^ Exponent  ^ Mantissa  ^ The smallest                             ^ The largest                              ^
-| Single (32 bit)  | 8 bits    | 23 bits   | {{ :en:multiasm:cs:min_fp_32.png?200 |}}  | 10(38.53)    |
+| Single (32 bit)  | 8 bits    | 23 bits   | {{ :en:multiasm:cs:min_fp_32.png?105 }}  | {{ :en:multiasm:cs:max_fp_32.png?100 }}  |
-| Double (64 bit)  | 11 bits   | 52 bits   | 10(-323.3)                                | 10(308.3)    |
+| Double (64 bit)  | 11 bits   | 52 bits   | {{ :en:multiasm:cs:min_fp_64.png?120 }}  | {{ :en:multiasm:cs:max_fp_64.png?100 }}  |
 </table>
+The most common representation for real numbers on computers is standardised in the document IEEE Standard 754. There are two modifications implemented which make the calculations easier for computers.
+  * The Biased exponent
+  * The Normalised Mantissa
+Biased exponent means that the bias value is added to the real exponent value. This results with all positive exponents which makes it easier to compare numbers.
+The normalised mantissa is adjusted to have only one bit of the value "1" to the left of the decimal. It requires an appropriate exponent adjustment.
 ===== Texts =====

en/multiasm/cs/chapter_3_11.1736358598.txt.gz · Last modified: 2025/01/08 17:49 by ktokarz