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    <section class="sect1" title="Numbers" epub:type="subchapter" id="Tutorial---Numbers">
      <div class="titlepage">
        <div>
          <div>
            <h2 class="title" style="clear: both">Numbers</h2>
          </div>
        </div>
      </div>
      <h3 id="idm139667880611536">Exact integers and fractions</h3>
      <p>Kawa has the usual syntax for decimal integers.
Addition, subtraction, and multiplication
are written using the usual <code class="literal">+</code>,
<code class="literal">-</code>, and  <code class="literal">*</code>,
but these are all prefix functions that take a variable number of arguments:
</p>
      <pre class="screen">(+ 1 2 3) ⇒ 6
(- 10 3 4) ⇒ (- (- 10 3) 4)  ⇒ 3
(* 2 -6)  ⇒ -12
</pre>
      <p>Kawa has arbitrary-precision integers.
</p>
      <p>Let us implement the <a class="ulink" href="http://en.wikipedia.org/wiki/Factorial" target="_top">factorial</a> function.
Type in the following (we’ll look at the syntax shortly):
</p>
      <pre class="screen"><span class="prompt">#|kawa:1|# </span><strong class="userinput"><code>(define (factorial x)</code></strong>
<span class="prompt">#|(---:2|# </span><strong class="userinput"><code>  (if (&lt; x 1) 1</code></strong>
<span class="prompt">#|(---:3|# </span><strong class="userinput"><code>    (* x (factorial (- x 1)))))</code></strong>
</pre>
      <p>(The prompt changes to indicate a continuation line.)
This binds the name <code class="literal">factorial</code>
to a new function, with formal parameter <code class="literal">x</code>.
This new function is immediately compiled to Java bytecodes,
and later a JIT compiler may compile it to native code.
</p>
      <p>A few tests:
</p>
      <pre class="screen"><span class="prompt">#|kawa:4|# </span><strong class="userinput"><code>(list (factorial 3) (factorial 4))</code></strong>
(6 24)
<span class="prompt">#|kawa:5|# </span><strong class="userinput"><code>(factorial 30)</code></strong>
265252859812191058636308480000000
</pre>
      <h3 id="idm139667880601184">Floating-point real numbers</h3>
      <p>Given what was said above about being able to add, subtract and multiply integers,
the following may be unexpected:
</p>
      <pre class="screen">#|kawa:1|# (/ 2 3)
2/3
#|kawa:2|# (+ (/ 1 3) (/ 2 3))
1
</pre>
      <p>In many languages, dividing two integers, as 2/3, would result in 0.  At best,
the result would be a floating point number, similar to 0.666667.  Instead,
Kawa has a <span class="emphasis"><em>rational</em></span> number type, which holds the results of divisions
<span class="emphasis"><em>exactly</em></span>, as a proper fraction.  Hence, adding one third to two thirds
will always result in exactly one.
</p>
      <p>Floating-point real numbers are known in Kawa as <span class="emphasis"><em>inexact</em></span> numbers, as
they cannot be stored exactly.  Consider:
</p>
      <pre class="screen">#|kawa:3|# (exact? 2/3)
#t
#|kawa:4|# (exact? 0.33333333)
#f
#|kawa:5|# (exact-&gt;inexact 2/3)
0.6666666666666666
</pre>
      <p>The first two examples check numbers for being <code class="literal">exact?</code>; there is a
corresponding <code class="literal">inexact?</code> test.  The last shows how an exact number can be
converted to an inexact form.
</p>
      <p>Numbers are converted between exact and inexact versions when required
within operations or procedures:
</p>
      <pre class="screen">#|kawa:6|# (+ 0.33333333 2/3)
0.9999999966666666
#|kawa:7|# (inexact? (+ 0.33333333 2/3))
#t
#|kawa:8|# (sin 2/3)
0.618369803069737
</pre>
      <h3 id="idm139667880593984">Complex numbers</h3>
      <p>A <span class="emphasis"><em>complex</em></span> number is made from two parts: a <span class="emphasis"><em>real</em></span> part and an
<span class="emphasis"><em>imaginary</em></span> part.  They are written <code class="literal">2+3i</code>.  A complex number can
be manipulated just like other numbers:
</p>
      <pre class="screen">#|kawa:9|# (+ 2+3i 5+2i)
7+5i
#|kawa:10|# (* 2+3i 4-3i)
17+6i
#|kawa:11|# (integer? (+ 2+3i -3i))
#t
</pre>
      <p>Notice how in the last example the result is an integer, which Kawa recognises.
</p>
      <p>Kawa also includes <a class="link" href="Quaternions.xhtml" title="Quaternions">quaternion</a> numbers.
</p>
      <h3 id="idm139667880588928">Units and dimensions</h3>
      <p>In many applications, numbers have a <span class="emphasis"><em>unit</em></span>.  For example, 5 might be
a number of dollar bills, a weight on a scale, or a speed.  Kawa enables us
to represent numbers as <span class="emphasis"><em>quantities</em></span>: numbers along with their unit.
For example, with weight, we might measure weight in pounds and ounces,
where an ounce is 1/16 of a pound.
</p>
      <p>Using Kawa, we can define units for our weight measurements, and specify the 
units along with numbers:
</p>
      <pre class="screen">#|kawa:12|# (define-base-unit pound "Weight")
#|kawa:13|# (define-unit ounce 0.0625pound)
#|kawa:14|# 3pound
3.0pound
#|kawa:15|# (+ 1pound 5ounce)
1.3125pound
</pre>
      <p>In this example we define a base unit, the pound, and a unit based on it, the
ounce, which is valued at 0.0625 pounds (one sixteenth).  Numbers can then be
written along with their unit (making them quantities).  Arithmetic is possible 
with quantities, as shown in the last line, and Kawa will do the smart thing
when combining units.  In this case, 1 pound and 5 ounces is combined to make
1.3125 pounds.
</p>
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