blocks|key|161338|text|虽然所有其他答案都是好的，但仍然缺少一件事：|type|unstyled|depth|inlineStyleRanges|entityRanges|data|161339|表示无理数是不可能的(例如，sqrt(2)**，**+log(3)**，+π等)准确地说！**|offset|length|style|BOLD|CODE|161340|这实际上就是它们被称为非理性的原因。世界上任何数量的比特存储都不足以容纳它们中的一个。只有符号算术才能保持它们的精度。|161341|但是，如果您将您的数学需求限制为有理数，那么精度问题就变得可控了。您需要存储一对(可能非常大的)整数a和b，以保存分数a/b表示的数字。你所有的算术运算都必须像高中数学一样做分数运算(例如a/b+*+c/d+=+ac/bd)。|161342|但当然，当涉及到pi、sqrt、log、sin等时，你仍然会遇到同样的麻烦。|161343|TL;DR|161344|对于硬件加速算法，只能表示有限数量的有理数。每个不可表示的数字都是近似的。有些数字(即无理数)永远不能表示，无论系统是什么。|161345|entityMap^0|0|E|9|R|8|E|7|R|6|0|0|1E|1|1G|1|1N|3|2M|H|0|8|2|B|4|G|3|K|3|0|0|5|0|0^^$0|@$1|2|3|4|5|6|7|U|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|V|8|@$D|W|E|X|F|G]|$D|Y|E|Z|F|G]|$D|10|E|11|F|H]|$D|12|E|13|F|H]]|9|@]|A|$]]|$1|I|3|J|5|6|7|14|8|@]|9|@]|A|$]]|$1|K|3|L|5|6|7|15|8|@$D|16|E|17|F|H]|$D|18|E|19|F|H]|$D|1A|E|1B|F|H]|$D|1C|E|1D|F|H]]|9|@]|A|$]]|$1|M|3|N|5|6|7|1E|8|@$D|1F|E|1G|F|H]|$D|1H|E|1I|F|H]|$D|1J|E|1K|F|H]|$D|1L|E|1M|F|H]]|9|@]|A|$]]|$1|O|3|P|5|6|7|1N|8|@$D|1O|E|1P|F|G]]|9|@]|A|$]]|$1|Q|3|R|5|6|7|1Q|8|@]|9|@]|A|$]]|$1|S|3|-4|5|6|7|1R|8|@]|9|@]|A|$]]]|T|$]]

While all of the other answers are good there is still one thing missing:

It is impossible to represent irrational numbers (e.g. π, <code>sqrt(2)</code>, <code>log(3)</code>, etc.) precisely!

And that actually is why they are called irrational. No amount of bit storage in the world would be enough to hold even one of them. Only symbolic arithmetic is able to preserve their precision.

Although if you would limit your math needs to rational numbers only the problem of precision becomes manageable. You would need to store a pair of (possibly very big) integers <code>a</code> and <code>b</code> to hold the number represented by the fraction <code>a/b</code>. All your arithmetic would have to be done on fractions just like in highschool math (e.g. <code>a/b * c/d = ac/bd</code>).

But of course you would still run into the same kind of trouble when <code>pi</code>, <code>sqrt</code>, <code>log</code>, <code>sin</code>, etc. are involved.

TL;DR

For hardware accelerated arithmetic only a limited amount of rational numbers can be represented. Every not-representable number is approximated. Some numbers (i.e. irrational) can never be represented no matter the system.

blocks|key|2754685|text|有无限多的实数(如此之多以至于你不能枚举它们)，也有无限多的有理数(可以枚举它们)。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|2754686|浮点表示是有限的(就像计算机中的任何东西一样)，因此不可避免地会有许多数字无法表示。特别是，64位只允许您区分18,446,744,073,709,551,616个不同的值(与无穷大相比，这算不了什么)。对于标准约定，9.2不在其中。对于某些整数m和e，可以是m.2%5Ee的形式。|2754687|2754688|您可能会想出一个不同的计数系统，例如，以10为基数，其中9.2将具有精确的表示。但其他数字，比如1/3，仍然不可能表示出来。|2754689|2754690|还要注意，双精度浮点数是非常精确的。它们可以用多达15位的精确数字来表示非常宽范围内的任何数字。对于日常生活计算，4或5位数就足够了。你永远不会真的需要这15个，除非你想计算你生命中的每一毫秒。|2754691|entityMap^0|0|0|0|0|0|0^^$0|@$1|2|3|4|5|6|7|L|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|M|8|@]|9|@]|A|$]]|$1|D|3|-4|5|6|7|N|8|@]|9|@]|A|$]]|$1|E|3|F|5|6|7|O|8|@]|9|@]|A|$]]|$1|G|3|-4|5|6|7|P|8|@]|9|@]|A|$]]|$1|H|3|I|5|6|7|Q|8|@]|9|@]|A|$]]|$1|J|3|-4|5|6|7|R|8|@]|9|@]|A|$]]]|K|$]]

There are infinitely many real numbers (so many that you can't enumerate them), and there are infinitely many rational numbers (it is possible to enumerate them).

The floating-point representation is a finite one (like anything in a computer) so unavoidably many many many numbers are impossible to represent. In particular, 64 bits only allow you to distinguish among only 18,446,744,073,709,551,616 different values (which is nothing compared to infinity). With the standard convention, 9.2 is not one of them. Those that can are of the form m.2^e for some integers m and e.

<hr>

You might come up with a different numeration system, 10 based for instance, where 9.2 would have an exact representation. But other numbers, say 1/3, would still be impossible to represent.

<hr>

Also note that double-precision floating-points numbers are extremely accurate. They can represent any number in a very wide range with as much as 15 exact digits. For daily life computations, 4 or 5 digits are more than enough. You will never really need those 15, unless you want to count every millisecond of your lifetime.

Why do some numbers lose accuracy when stored as floating point numbers?

For example, the decimal number <code>9.2</code> can be expressed exactly as a ratio of two decimal integers (<code>92/10</code>), both of which can be expressed exactly in binary (<code>0b1011100/0b1010</code>). However, the same ratio stored as a floating point number is never exactly equal to <code>9.2</code>:

<pre><code>32-bit "single precision" float: 9.19999980926513671875
64-bit "double precision" float: 9.199999999999999289457264239899814128875732421875
</code></pre>

How can such an apparently simple number be "too big" to express in 64 bits of memory?

Why are floating point numbers inaccurate?

为什么有些数字在存储为浮点数时会失去准确性？例如，十进制数9.2可以精确地表示为两个十进制整数(92/10)的比率，这两个整数都可以精确地表示为二进制(0b1011100/0b1010)。但是，存储为浮点数的相同比率永远不会完全等于9.232-bit "single precision" float: 9.199999...

问为什么浮点数是不准确的？
EN

社区

活动

资源

关于

腾讯云开发者

热门产品

热门推荐

更多推荐

问为什么浮点数是不准确的？EN

社区

活动

资源

关于

腾讯云开发者

热门产品

热门推荐

更多推荐

问为什么浮点数是不准确的？
EN