D provides fundamental data types for integers and floating-point constants. Arithmetic may only be performed on integers in D programs. Floating-point constants may be used to initialize data structures, but floating-point arithmetic is not permitted in D. D provides a 32-bit and 64-bit data model for use in writing programs. The data model used when executing your program is the native data model associated with the active operating system kernel. You can determine the native data model for your system using isainfo -b.
The names of the integer types and their sizes in each of the two data models are shown in the following table. Integers are always represented in twos-complement form in the native byte-encoding order of your system.
How big is an integer
In C#, all numeric data types store limited range of values. For example, Int32 data type can store integers from -2,147,483,648 to 2,147,483,647. The long (Int64) type can store integers from -9,223,372,036,854,775,808 to 9,223,372,036,854,775,807, and unsigned long has the limit of 0 to 18,446,744,073,709,551,615.
Convert a Python integer pylong to a C void pointer.If pylong cannot be converted, an OverflowError will be raised. Thisis only assured to produce a usable void pointer for values createdwith PyLong_FromVoidPtr().
For example, a 64-bit integer cannot be represented in JSON (since JavaScript and JSON support integers up to 2^53). Therefore, a 64-bit integer must be represented as a string in JSON requests/responses. So the type property will be set to "string", but the format property will be set to "int64" to indicate that it is a 64-bit integer.
Long is a data type in VBA used to store the numeric values. We know that integer also holds numeric values, but Long differs from integers as the data storage range is very big. In the case of the Long data type, we can hold decimal values too. So, it is a built-in data type.
With this VLDB property you can determine whether long integers are mapped to a BigInt data type when MicroStrategy creates tables in the database. A data mart is an example of a MicroStrategy feature that requires MicroStrategy to create tables in a database.
When long integers from databases are integrated into MicroStrategy, the Big Decimal data type is used to define the data in MicroStrategy. Long integers can be of various database data types such as Number, Decimal, and BigInt.
The BigInt data type in SQL Server is the 64-bit representation of an integer. It takes up 8 bytes of storage. It can range from -2^63 (-9,223,372,036,854,775,808) to 2^63 (9,223,372,036,854,775,807). Two raised to the power of sixty-three is about nine quintillion, a very big number. Imagine a number greater than the number of grains of sand on Earth, it is this big.
In retrospect, you are dealing with two limits. One limit for the SQL Server BigInt data type and one for the JavaScript Number data type. The Node environment will support BigInt integers up to the JavaScript Number limit. The JavaScript limit is about nine quadrillion, so lower than a BigInt. But, can I do better? Is there a way to use the full extent of the BigInt data type? Yes, by representing them as strings.
This approach does come with limitations. A JavaScript string type is not an integer. You cannot do any math on it. It is impractical to increment a number represented by a string by one, for example, especially if it is higher than 9223372036854775806.
Use the DECIMAL or NUMERIC data type to store values with a user-defined precision. The DECIMAL and NUMERIC keywords are interchangeable. In this document, decimal is the preferred term for this data type. The term numeric is used generically to refer to integer, decimal, and floating-point data types.
These rules are because DECIMAL values with 19 or fewer significant digits of precision are stored internally as 8-byte integers, while DECIMAL values with 20 to 38 significant digits of precision are stored as 16-byte integers.
This article explains a very simple (and inefficient) implementation of an (unsigned) big integer library. While there are many open-source implementations of big integer libraries freely available on the internet, nearly all of them lack an explanation how they work internally.Actually I could not find any useful basic information about the algorithms used to store big integers and implement arithmetic operations. Indeed, there are many theoretical articles and books available about so-called arbitrary precision arithmetic (e.g. in [5]), but they are quite complex to understand and not feasible as a practical starting point.So I thought why not trying myself writing a big integer library and share my knowledge with you :-).
A more comfortable way is creating unsigned big integer objects from unsigned integers like uint or ulong.Following algorithm describes one way to fulfill this task. Note that integer divisions are used, no floating point division.
A remarkable limitation in creating big integers from native unsigned integers is that only numbers in range of the supported unsigned integers can be created and not arbitrary long ones as initially intended.
Creating big integers from strings finally allows to create big integers with an insane number of digits - only the available memory limits the size of the numbers.Based on the fact that each character of the string represents a single digit, just iterate over each character of the string and convert it to a digit. This turns out surprisingly easy because the ASCII code of digit '0' is 0x30, '1' is 0x31 ... and '9' is 0x39. So by subtracting 0x30 from a single ASCII character, the decimal digit can be retrieved.
It might become necessary to create big integer objects from string in hexadecimal notion like e.g. "0x3A". This is a bit more complex than the decimal case in previous chapter, as here a single string character cannot necessarily be represented also as a single digit. For example, the single character 'C' stands for decimal number 12 which has two digits.At first, consider in general the conversion of hexadecimal number to a decimal number. A hexadecimal number can also be written in positional notation, just with base 16. As a numerical system with base 16 uses also 16 different digits, beside the digits in range [0, 9], the characters in range [A, F] are interpreted as digits 10, 11, .. , 15.
But stop! In third last line, multiplication and addition is used - for our big integer objects.Yes, that's right, currently these two arithmetic operations are required for the algorithms and have not been addressed yet. They will be discussed later in subsequent chapters.
Being able to create big integer objects (that consists actually only of an array of digits), now it's time to present operations on those objects. Be prepared to be confronted with primary school mathematics!
It's worth to remind at this point that an unsigned big integer library is implemented. Negative numbers are not supported, it must be ensured that the first number a is greater than the second number b, otherwise the result is meaningless.This implies that the number of digits of a is equal or greater than the number of digits of b. Further, the number of digits of the result is not greater than the number of digits of a.A possible implementation could look like following:
There is one important point here: For the product of two single digits, primitive types can be used as the maximum value could be 9 * 9 + 9 = 90. This is good news as we need division by 10 and modulo by 10 for this calculation.But note that the two factors a and b could have arbitrary length, thus also the result variable as well as the rowProduct variable must be big integer objects.But wait, for result and rowProduct variables, beside addition we need to multiply by 10n - and multiplication is not yet defined for big integer objects! So what to do here?Fortunately, there is a trick here:Multiplying a big integer object by 10n is the same as left shifting it by n positions to the left because base 10 is used. This is similar to left shifting a primitive integer i objects by n resulting in i * 2n because they are stored in base 2.
Division for arbitrary precision arithmetic numbers is by far the most complicated artihmetic operation. There are very complex algorithms described in literature which are hard to understand and which would require own tutorials itself to explain. As the goal of this article is to implement a working, but simple (at the cost of efficiency) big unsigned integer, here the probably simplest algorithm for division is presented. It's a recursive algorithm, but first start with the basics.Problem definition:The aim is to calculate the quotient and remainder of the division of the two numbers a (dividend) and b (divisor):The quotient q is defined as the result of division: q = a / b.The remainder r is defined as the result of the modulo operation: r = a % b.This can also be summarized and written as a = q * b + r.Algorithm:The algorithm distinguishes three cases.Case 1: If a The main idea of the algorithm is to recursively multiply the divisor b by 2 until the trivial case is reached.:a = q * b + r => a = q' * 2 * b + r'.Then travserse the taken recursive way back to the root and calculate calculate q' and r' each step to finally gain q and r.The next two cases handle the general cases. The distinction is based on the requirement that the remainder r is always smaller than the divisor b: Constraint: r Case 2: The doubling of the divisor b does not violate constraint r Assume from a = q * b + r, the step to a = q' * 2 * b + r' has been done and the values q' and r' have been calculated. Now the step back shall be taken to retrieve q and r. If r' is smaller than b, then the remainder remains the same and q is doubled: q = 2 * q' and r = r'. 2ff7e9595c
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