如果不能容纳足够多的有效数字，那么双精度数如何代表比小数更高的数字？

Question

I may very well have not the proper understanding of significant figures , but the book

C# 6.0 in a Nutshell by Joseph Albahari and Ben Albahari (O'Reilly).

Copyright 2016 Joseph Albahari and Ben Albahari, 978-1-491-92706-9.

provides the table below for comparing double and decimal :

Is it not counter-intuitive that, on the one hand, a double can hold a smaller quantity of significant figures , while on the other it can represent numbers way bigger than decimal, which can hold a higher quantity of significant figures ?

Answer 1

Imagine you were told you can store a value, but were given a limitation: You can only store 10 digits, 0-9 and a negative symbol. You can create the rules to decode the value, so you can store any value.

The first way you store things is simply as the value xxxxxxxxxx , meaning the number 123 is stored as 0000000123 . Simple to store and read. This is how an int works.

Now you decide you want to store fractional numbers, so you change the rules a bit. Now you store xxxxxxyyyy , where x is the integer portion and y is the fractional portion. So, 123.98 would be stored as 0001239800 . This is roughly how a Decimal value works. You can see the largest value I can store is 9999999999 , which translates to 999999.9999. This means I have a hard upper limit on the size of the value, but the number of the significant digits is large at 10.

There is a way to store larger values, and that's to store the x and y components for the formula in xxxxxxyyyy . So, to store 123.98, you need to store 01239800-2 , which I can calculate as . This means I can store much bigger numbers by changing 'y', but the number of significant digits is basically fixed at 6. This is basically how a double works.

Answer 2

The answer lies in the way that doubles are encoded. Rather than just being a direct binary representation of a number, they have 3 parts: sign, exponent, and fraction.

• The sign is obvious, it controls + or -.

• The fraction part is also obvious. It's binary fraction that represents a number in between 0 and 1.

• The exponent is where the magic happens. It signifies a scaling factor.

The final float calculation comes out to (-1)^$sign * (1 + $fraction) * 2 ^$exponent

This allows much higher values than a straight decimal number because of the exponent. There's a lot of reading out there on why this works and how to do addition and multiplication with these encoded numbers. Google around for "IEEE floating point format" or whatever topic you need. Hope that helps!

Answer 3

The Range has nothing to do with the precision. Double has a binary representation (base 2). Not all numbers can be represented exactly as we humans know them in the decimal format. Not to mention and accumulated rounding errors of addition and division. A larger range means a greater MAX VALUE and a smaller MIN VALUE than decimal.

Decimal on the other side is (base 10). It has a smaller range (smaller MAX VALUE and larger MIN VALUE). This has nothing to do with precision, since it is not represented using floating binary point representation , it can represent numbers more precisely and though is recommended for human-made numbers and calculations.