0是不是自然数？
<br /="/"/>今天看到新闻热评里对小学数学教材作了改动，把原来认为不是自然数的0规定为自然数。有关报道见内嵌网页。<div></div><div><br /="/"/></div><div>请问：</div><div>1.国际数学界对于0是否自然数是如何规定的？为什么？基于何种考虑？</div><div>2.这一基本概念的改动会否引起有关数学定理的变化？对基础数学研究有何影响？</div><div><br /="/"/></div><div><br /="/"/></div><div><br /="/"/></div><div><br /="/"/></div><br /="/"/><br /="/"/><iframe height="500" src="http://www.ddhw.com/viewheadlinenews.aspx?topic_id=1000&msg_id=119976" width="680"></iframe><br /="/"/><br /="/"/><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:5px;MARGIN-BOTTOM:10px"><br /="/"/><img /="/" src="/img/sign.gif"></img><a href="http://www.ddhw.org" target="_top"><span style="FONT-SIZE:12px;COLOR:#999999;font-weight:bold">www.ddhw.org</span></a>---<div style="width:450px;MARGIN-LEFT:10px"><div></div><p><input /="/" src="http://upload.ddhw.cn/image/2008/05/28/9889.jpg" style="CURSOR: default" type="image"/></p><p>据说这世界是彩色的？<div> </div><div></div></p></div></div>回复：0是不是自然数？
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>传统的自然数是不包括0，因为自然数就是用来数东西的，从1开始比较直观。</div><div>近代的说法包括0，这样在集合论里会方便一些。<span style="display:none;">www.ddhw.com</span></div><div>通常用下面的记号可以避免歧义，</div><div>N<sup>0={0,1,2...}</sup></div><div><sup>N*={1,2,3....}</sup></div><div>这个对数学定理没什么影响，没什么变化，这是个芝麻小问题，算不上是数学问题。</div><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>回复：0是不是自然数？
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div><div>俺的感觉也是把0放在自然数中会比较方便。从整数的近代定义来看，从0 （0对应于空集<strong><font color="#00cc00">{}</font></strong>，1对应于<font color="#ff0000">{</font><font color="#00cc00">{}</font><font color="#ff0000">}</font>，2对应于<font color="#ff9900">{</font><font color="#00cc00">{}</font>,<font color="#ff0000">{</font><font color="#00cc00">{}</font><font color="#ff0000">}</font><font color="#ff9900">}</font>，......）出发也很合理、自然。事物都在变化，科学也在发展，教科书与时俱进是理所当然的。</div></div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>回复：回复：0是不是自然数？
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/>报道最后援引的说法称把0规定为自然数是国际标准。请问是否存在这样的国际标准？在国际上进行数学讨论的时候，在0是否自然数会对讨论结果有影响的时候，是特意说明还是有一个默认的定义？<div></div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:5px;MARGIN-BOTTOM:10px"><br /="/"/><img /="/" src="/img/sign.gif"></img><a href="http://www.ddhw.org" target="_top"><span style="FONT-SIZE:12px;COLOR:#999999;font-weight:bold">www.ddhw.org</span></a>---<div style="width:450px;MARGIN-LEFT:10px"><div></div><p><input /="/" src="http://upload.ddhw.cn/image/2008/05/28/9889.jpg" style="CURSOR: default" type="image"/></p><p>据说这世界是彩色的？<div> </div><div></div></p></div></div></td></tr></table>看看维基说什么？[:-D]
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div><p>In <a href="http://en.wikipedia.org/wiki/Mathematics" title="Mathematics"><u>mathematics</u></a>, the <b>natural numbers</b> are the ordinary whole numbers used for <a href="http://en.wikipedia.org/wiki/Counting" title="Counting"><u>counting</u></a> ("there are 6 coins on the table") and <a href="http://en.wikipedia.org/wiki/Total_order" title="Total order"><u>ordering</u></a> ("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see <a href="http://en.wikipedia.org/wiki/English_numerals" title="English numerals"><u>English numerals</u></a>). A later notion is that of a <a href="http://en.wikipedia.org/wiki/Nominal_number" title="Nominal number"><u>nominal number</u></a>, which is used only for naming.<span style="display:none;">www.ddhw.com</span></p><p>Properties of the natural numbers related to <a href="http://en.wikipedia.org/wiki/Divisibility" title="Divisibility"><u>divisibility</u></a>, such as the distribution of <a href="http://en.wikipedia.org/wiki/Prime_number" title="Prime number"><u>prime numbers</u></a>, are studied in <a href="http://en.wikipedia.org/wiki/Number_theory" title="Number theory"><u>number theory</u></a>. Problems concerning counting and ordering, such as <a href="http://en.wikipedia.org/wiki/Partition_(number_theory)" title="Partition (number theory)"><u>partition</u></a> <a href="http://en.wikipedia.org/wiki/Enumeration" title="Enumeration"><u>enumeration</u></a>, are studied in <a href="http://en.wikipedia.org/wiki/Combinatorics" title="Combinatorics"><u>combinatorics</u></a>.<span style="display:none;">www.ddhw.com</span></p><p><font color="#cc0000">There is no universal agreement about whether to include zero in the </font><a href="http://en.wikipedia.org/wiki/Set_(mathematics)" title="Set (mathematics)"><u><font color="#cc0000">set</font></u></a><font color="#cc0000"> of natural numbers</font>: some define the natural numbers to be the <b><a href="http://en.wikipedia.org/wiki/Positive_number" title="Positive number"><u>positive</u></a> <a href="http://en.wikipedia.org/wiki/Integer" title="Integer"><u>integers</u></a></b> <span>{<a href="http://en.wikipedia.org/wiki/1_(number)" title="1 (number)"><u>1</u></a>, <a href="http://en.wikipedia.org/wiki/2_(number)" title="2 (number)"><u>2</u></a>, <a href="http://en.wikipedia.org/wiki/3_(number)" title="3 (number)"><u>3</u></a>, ...</span>}, while for others the term designates the <b><a href="http://en.wikipedia.org/wiki/Non-negative" title="Non-negative"><u>non-negative</u></a> integers</b> <span>{<a href="http://en.wikipedia.org/wiki/0_(number)" title="0 (number)"><u>0</u></a>, 1, 2, 3, ...</span>}. The former definition is the traditional one, with the latter definition first appearing in the 19th century. Some authors use the term "natural number" to exclude zero and "<a href="http://en.wikipedia.org/wiki/Whole_number" title="Whole number"><u>whole number</u></a>" to include it; others use "whole number" in a way that excludes zero, or in a way that includes both zero and the negative integers.</p></div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>
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