Interleave sequence

<h2 id="properties">Properties</h2>
<ul><li>The interleave sequence 
 
 
 
 (
 
 z
 
 i
 
 
 )
 
 
 {\displaystyle (z_{i})}
 
 is <a href="/facts/Convergent_sequence/Ktge2RwL">convergent</a> <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the sequences 
 
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 {\displaystyle (x_{i})}
 
 and 
 
 
 
 (
 
 y
 
 i
 
 
 )
 
 
 {\displaystyle (y_{i})}
 
 are convergent and have the same <a href="/facts/Limit_of_a_sequence/Ktge2RwL">limit</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li>
<li>Consider two <a href="/facts/Real_number/R02gw5Pb">real numbers</a> a and b greater than zero and smaller than 1. One can interleave the sequences of digits of a and b, which will determine a third number c, also greater than zero and smaller than 1. In this way one obtains an <a href="/facts/Injective_function/5ES6tqf5">injection</a> from the square (0, 1) × (0, 1) to the <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> (0, 1). Different <a href="/facts/Radix/J5RGWRJF">radixes</a> give rise to different injections; the one for the <a href="/facts/Binary_number/a6JDSRPs">binary numbers</a> is called the <a href="/facts/Z-order_curve/0QMjw8nF">Z-order curve</a> or Morton code.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li></ul>

This article incorporates material from Interleave sequence on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Strichartz, Robert S. (2000), The Way of Analysis, Jones & Bartlett Learning, p. 78, ISBN 9780763714970. <a href="9780763714970" target="_blank">9780763714970</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Mamoulis, Nikos (2012), Spatial Data Management, Synthesis lectures on data management, vol. 21, Morgan & Claypool Publishers, pp. 22–23, ISBN 9781608458325. <a href="9781608458325" target="_blank">9781608458325</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Interleave sequence open-in-new

Interleave sequence