{"id":42,"date":"2020-11-25T21:01:07","date_gmt":"2020-11-26T02:01:07","guid":{"rendered":"https:\/\/www.alpharithms.com\/?p=42"},"modified":"2021-07-20T22:45:12","modified_gmt":"2021-07-21T02:45:12","slug":"numerical-representation-in-computing","status":"publish","type":"post","link":"https:\/\/www.alpharithms.com\/numerical-representation-in-computing-070121\/","title":{"rendered":"Numerical Representation in Computing"},"content":{"rendered":"

Computer number formats are at the core of computer science. While not as heavy in theory as some other concepts, the practical implications of how numbers get represented in computers has drastic consequences in computing.<\/p>\n

Binary is often easily recognized, even by non-computer scientists, as it forms the basis of most modern computing numerical systems. However, numerical representation is a bit more complicated than just converting things to 1's and 0's<\/code>. Ther are octal, hexadecimal, floating points, doubles, longs, shorts\u2014the list goes on. At the heart of it all, however, are ones and zeroes.<\/p>\n

Binary Notation<\/h2>\n

Binary notation is a system by which numbers are represented using only 1’s and 0’s. This is also referred to as Base2 notation, as opposed to Base10 that everyone is familiar with. Conceptually, binary means counting in powers of two whereas base10 notation means counting in powers of 10. It’s a simple concept but can give rise to some complicated situations.<\/p>\n

\"binary
Binary notation uses base2 notation which calculates numbers based on powers of 2<\/figcaption><\/figure>\n

Signed vs. Unsigned<\/h3>\n

Computer scientists represent numbers in many ways. Two basic variations are signed <\/em>and unsigned<\/em> notations. These are simple accommodations to allow higher maximum values when only positive numbers are required for calculations. Consider the following:<\/p>\n

Signed Notation<\/strong> is the formal terminology that means both positive and negative. The word “signed” simply refers to the plus or minus\u00a0sign<\/em> commonly used to denote positive and negative values.<\/p>\n

Unsigned Notation<\/strong> is the formal terminology that means only positive values. This means of representation is used to accommodate higher maximum values when it’s known that negative values will not need to be represented.<\/p>\n

\"signed
The term “signed” means a number can be either positive or negative<\/figcaption><\/figure>\n

When a numeric representation system needs to support both positive\u00a0and<\/em> negative values, the maximum value possible within the system is effectively halved. In most cases, the maximum value becomes max \/ 2 - 1<\/code>.<\/p>\n

In many cases, the absolute value of the maximum value equals the absolute value of the minimum value, minus one. The issue arises because 0 must be accommodated, such that either the smallest or largest value must be dropped to make room, while still fitting into the available slots.<\/p>\n

\"signed
Supporting signed values effectively halves the largest possible value.<\/figcaption><\/figure>\n

Range of Numbers<\/h3>\n

The range of numbers able to be represented by computers is directly related to the size of available memory. For example, 32-bit memory architecture provides 32 bits with which numbers can be represented (4 bytes) and 64-bit architecture provides 64. Simple enough, right?<\/p>\n

Given what we now know about binary representation, we can calculate the largest number a 32-bit system can represent in binary would be 32-1’s or 232<\/sup> which is 4,294,967,296<\/code>.\u00a0 However, this isn’t correct yet\u2014we need to be able to accommodate 0 in our numbers.<\/p>\n

To make room for zero, computer scientists simply subtract 1 from the max value. This results in a range of 0 to 4,294,967,295<\/code>. An abbreviated example of this can be seen with a 4-bit memory architecture in the following illustration:<\/p>\n

\"twos
The maximum value in unsigned binary notation can be calculated using the (2^n)-1 notation<\/figcaption><\/figure>\n

This is the standard range for an unsigned\u00a0<\/em>integer in languages like C++ (R<\/a>). The term\u00a0unsigned<\/em> means that all numbers are assumed positive. This is represented slightly differently for signed values.<\/p>\n

For signed <\/i>values\u2014numbers having a positive or negative sign\u2014the range of numbers is still the same amount, but it’s split between positive and negative values. The 0\u2014232<\/sup> -1 range becomes -231<\/sup> \u2014 231<\/sup> -1 or -2,147,483,648 \u2014 2,147,483,647<\/code>. Again, note the reduction of the max value by 1 to accommodate 0.<\/p>\n

The general form for calculating the range in numbers for a signed number, given the maximum number of available bits (M) is as follows: -2m-1<\/sup> \u2014 2m-1<\/sup> – 1.<\/p>\n

Where 32 bits is the max number of bits available, our max value becomes 31. So, for example, in modern systems where 64-bit memory allows 263<\/sup> bit representation, the range of possible\u00a0signed<\/em> values becomes -9,223,372,036,854,775,808 \u20149,223,372,036,854,775,807<\/code>.<\/p>\n

Work in Progress<\/h2>\n

This post is a work in progress and currently represents an incomplete discussion. Future points of focus include the following areas of numerical representation:<\/p>\n