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425 lines
20 KiB
HTML
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HTML
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<title>The Evolution of Numbers</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">The Evolution of Numbers</h1>
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<p class="center"><img src="images/evolution-of-numbers.svg" alt="The Evolution of Numbers" height="131" width="643"></p>
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<p>I want to take you on an adventure ...</p>
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<p class="indent50px"><b>... an adventure through the world of numbers.</b></p>
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<p>Let us start at the beginning:</p>
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<div class="indent50px">
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<p class="larger"><span class="large">Q: </span> What is the simplest idea of a number?</p>
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<p class="larger"><span class="large">A: </span> Something to <b>count</b> with!</p>
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</div>
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<h2>The Counting Numbers</h2>
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<p>We can use numbers to <b>count</b>: 1, 2, 3, 4, etc</p>
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<p>Humans have been using numbers to count with for thousands of years. It is a very natural thing to do.</p>
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<ul>
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<li>You can have "<b>3</b> friends",</li>
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<li>a field can have "<b>6</b> cows"</li>
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<li>and so on.</li>
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</ul>
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<p>So we have:</p>
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<div class="def">
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<p>Counting Numbers: {1, 2, 3, ...}</p>
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</div>
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<p>And the "Counting Numbers" satisfied people for a long time.</p>
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<h2>Zero</h2>
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<p>The idea of <b><a href="zero.html">zero</a></b>, though natural to us now, was not natural to early humans ... if there is nothing to count, how can we count it?</p>
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<p class="center"><b>Example: we can count dogs, but we can't count an empty space:</b></p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="../sets/images/saluki.jpg" alt="2 dogs" height="112" width="250"></td>
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<td> </td>
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<td><img src="../sets/images/saluki-none.jpg" alt="no dogs" height="112" width="250"></td>
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</tr>
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<tr>
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<th align="center">Two Dogs</th>
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<th align="center"> </th>
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<th align="center">Zero Dogs? Zero Cats?</th>
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</tr>
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</tbody></table>
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<p>An empty patch of grass is just an empty patch of grass!</p>
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<h3>Placeholder</h3>
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<p>But about 3,000 years ago people needed to tell the difference between numbers like <b>4</b> and <b>40.</b> Without the zero they look the same!</p>
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<p>So they used a "placeholder", a space or special symbol, to show "there are no digits here"</p>
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<div class="center80">
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<p class="def" style="font-size: 2rem; float:left; margin: 0 10px 5px 0; text-align:center; ">5 2</p>
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<p>So "5 2" meant "502" (5 hundreds, nothing for the tens, and 2 units)</p>
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</div>
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<h3>Number</h3>
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<p>The idea of zero had begun, but it wasn't for another thousand years or so that people started thinking of it as an actual <b>number</b>.</p>
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<p>But now we can think</p>
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<p class="center"><i>"I had 3 oranges, then I ate the 3 oranges, now I have <b>zero</b> oranges...!"</i></p>
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<h2>The Whole Numbers</h2>
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<p>So, let us add zero to the counting numbers to make <b>a new set of numbers.</b></p>
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<p>But we need a new name, and that name is "Whole Numbers":</p>
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<div class="def">
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<p><a href="../whole-numbers.html">Whole Numbers</a>: {0, 1, 2, 3, ...}</p>
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</div>
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<p class="center"><img src="../images/number-line-positive.gif" alt="whole number line" height="69" width="426"></p>
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<h2>The Natural Numbers</h2>
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<p>You may also hear the term "<b>Natural Numbers</b>" ... which can mean:</p>
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<ul>
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<li>the "Counting Numbers": {1, 2, 3, ...}</li>
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<li><b>or</b> the "Whole Numbers": {0, 1, 2, 3, ...}</li>
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</ul>
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<p>depending on the subject. I guess they disagree on whether zero is "natural" or not.</p>
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<h2>Negative Numbers</h2>
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<p>But the history of mathematics is all about people asking questions, and seeking the answers!</p>
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<p>One of the good questions to ask is</p>
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<p class="center larger">"if we can go one way, can we go the <i>opposite</i> way?"</p>
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<p>We can count forwards: 1, 2, 3, 4, ...</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:center;">
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<p>... but what if we count backwards:</p>
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<p>3, 2, 1, 0, ... <b>what happens next?</b></p>
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</td>
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<td style="text-align:center;"> </td>
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<td><img src="images/number-line-below-zero.gif" alt="number line below zero" height="88" width="215"></td>
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</tr>
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</tbody></table>
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<p>The answer is: we get <span class="large">negative numbers:</span></p>
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<p class="center"><img src="images/number-line.svg" alt="number line" height="47" width="595"></p>
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<p class="center">Now we can go forwards and backwards as far as we want</p>
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<h3>But how can a number be "negative"?</h3>
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<p>By simply being less than zero.</p>
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<div class="simple">
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<table align="center" width="75%" border="0">
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<tbody>
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<tr>
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<td>
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<a href="../measure/thermometer.html"><img src="../measure/images/thermometer-thumb.gif" alt="thermometer" height="141" width="67"></a>
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</td>
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<td>
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<p>A simple example is <a href="../temperature-conversion.html">temperature</a>.</p>
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<p>We define zero degrees Celsius (<b>0° C</b>) to be when water freezes ... but if we get colder we need negative temperatures.</p>
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<p>So <b>−20° C</b> is 20° below Zero.</p>
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</td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/two-cows.jpg" alt="minus one cow" height="93" width="201"></p>
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<h3>Negative Cows?</h3>
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<p>And in theory we can have a negative cow!</p>
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<p>Think about this ...If you had just <b>sold two bulls</b>, but can only <b>find one</b> to hand over to the new owner... you actually <b>have minus one bull</b> ... you are in debt one bull!</p>
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<p>So negative numbers exist, and we're going to need a new set of numbers to include them ...</p>
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<h2>Integers</h2>
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<p>If we include the negative numbers with the whole numbers, we have a <b>new set of numbers</b> that are called <b>integers</b></p>
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<div class="def">
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<p>Integers: {..., −3, −2, −1, 0, 1, 2, 3, ...}</p>
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</div>
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<p>The Integers include zero, the counting numbers, and the negative of the counting numbers, to make a list of numbers that stretch in either direction indefinitely.</p>
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<p class="center">Try it yourself (click on the line):</p>
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<div class="script" style="height: 150px;">
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images/number-line.js?mode=int
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</div>
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<h2>Fractions</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/orange-halves.jpg" alt="orange halves" height="62" width="139"></p>
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<p>If you have one orange and want to share it with someone, you need to cut it in half.</p>
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<p>You have just invented a new type of number!</p>
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<p class="center"><b><i>You took a number (1) and divided by another number (2) to come up with half (1/2)</i></b></p>
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<p>The same thing happens when we have four biscuits (4) and want to share them among three people (3) ... they get (4/3) biscuits each.</p>
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<p>A new type of number, and a new name:</p>
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<h2>Rational Numbers</h2>
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<p>Any number that can be written as a fraction is called a Rational Number.</p>
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<p>So, if "p" and "q" are integers (remember we talked about integers), then p/q is a rational number.</p>
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<div class="example">
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<p>Example: If <b>p</b> is 3 and <b>q</b> is 2, then:</p>
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<p class="center"><b>p/q</b> = 3/2 = <b>1.5</b> is a rational number</p>
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</div>
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<p>The only time this doesn't work is when <b>q</b> is zero, because <a href="dividing-by-zero.html">dividing by zero</a> is undefined.</p>
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<div class="def">
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<p><a href="../rational-numbers.html">Rational Numbers</a>: {p/q : p and q are integers, q is not zero}</p>
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</div>
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<p>So half (<b>½</b>) is a rational number.</p>
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<p>And <b>2</b> is a rational number also, because we could write it as <b>2/1</b></p>
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<p>So, Rational Numbers include:</p>
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<ul>
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<li>all the <b>integers</b></li>
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<li>and all <b>fractions</b>.</li>
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</ul>
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<p>And also any number like 13.3168980325 is rational:</p>
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<p class="center">13.3168980325 = <span class="intbl"><em>133,168,980,325</em><strong>10,000,000,000</strong></span></p>
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<p>That seems to include all possible numbers, right?</p>
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<h2>But There Is More</h2>
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<p>People didn't stop asking the questions ...and here is one that caused a lot of fuss during the time of Pythagoras:</p>
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<div class="center80">
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/square-root-2.gif" alt="square root 2" height="115" width="102"> </p>
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<p>
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When we draw a square (of size "1"), what is the distance across the diagonal?
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</p>
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</div>
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<p>The answer is the <b><a href="../square-root.html">square root</a> of 2</b>, which is<b> 1.4142135623730950...(etc)</b></p>
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<p>But it is not a number like 3, or five-thirds, or anything like that ...</p>
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<div class="indent50px">
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<p>... in fact we <b>cannot</b> answer that question using a ratio of two integers</p>
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<p class="center larger">square root of 2 ≠ p/q</p>
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<p>... and so it is <b>not a rational number</b> <i>(read more <a href="irrational-finding.html">here</a>)</i></p>
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</div>
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<p>Wow! There are numbers that are NOT rational numbers! What do we call them?</p>
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<p class="center larger">What is "Not Rational" ...?<b> Irrational !</b></p>
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<h2>Irrational Numbers</h2>
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<p>So, the <b>square root of 2</b> (√2) is an <a href="../irrational-numbers.html">irrational</a> number. It is called irrational because it is not rational (can't be made using a simple ratio of integers). It isn't crazy or anything, just not rational.</p>
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<p>And we know there are many more irrational numbers. <a href="pi.html">Pi</a> (<span class="times">π</span>) is a famous one.</p>
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<h3>Useful</h3>
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<p>So irrational numbers are useful. We need them to</p>
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<ul>
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<li>find the diagonal distance across some squares,</li>
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<li>to work out lots of calculations with circles (using <span class="times">π</span>),</li>
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<li>and more,</li>
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</ul>
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<p>So we really should include them.</p>
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<p>And so, we introduce a new set of numbers ...</p>
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<h2>Real Numbers</h2>
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<p>That's right, another name!</p>
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<p>Real Numbers include:</p>
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<ul>
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<li>the rational numbers, and</li>
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<li>the irrational numbers</li>
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</ul>
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<div class="def">
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<p>Real Numbers: {x : x is a rational or an irrational number}</p>
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</div>
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<p>In fact a Real Number can be thought of as <b>any point</b> anywhere on the number line:</p>
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<div class="script" style="height: 150px;">
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images/number-line.js?mode=real
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</div>
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<p class="center"><i>This only shows a few decimal places (it is just a simple computer)<br>
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but Real Numbers can have <b>lots more decimal places</b>!</i></p>
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<p><b>Any</b> point <b>Anywhere</b> on the number line, that is surely enough numbers!</p>
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<p>But there is one more number which has turned out to be very useful. And once again, it came from a question.</p>
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<h2>Imagine ...</h2>
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<p>The question is:</p>
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<p class="center larger">"is there a <b>square root</b> of<b> minus one</b>?"</p>
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<p>In other words, <b>what can we multiply by itself to get −1</b>?</p>
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<p>Think about this: if we multiply any number by itself we can't get a negative result:</p>
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<ul>
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<li>1×1 = <b>1</b>,</li>
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<li>and also (−1)×(−1) = <b>1</b> (because a <a href="../multiplying-negatives.html">negative times a negative gives a positive</a>)</li>
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</ul>
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<p>So what number, when multiplied by itself, results in <b>−1</b>?</p>
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<p>This is normally not possible, but ...</p>
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<p class="center larger">"if you can imagine it, then you can play with it"</p>
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<p>So, ...</p>
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<h2>Imaginary Numbers</h2>
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<table width="100%" border="0">
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<tbody>
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<tr>
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<td><img src="images/imaginary-square-root.svg" alt="square root of minus one" height="44" width="135"></td>
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<td>
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<p>... let us just <b>imagine</b> that the square root of minus one <b>exists</b>.</p>
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<p>We can even give it a special symbol: the letter <span class="large">i</span></p>
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</td>
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</tr>
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</tbody></table>
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<p>And we can <b>use it</b> to answer questions:</p>
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<div class="example">
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<p>Example: what is the square root of −9 ?</p>
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||
<p>Answer: √(−9) = √(9 × −1) = √(9) × √(−1) = 3 × √(−1) = 3<span class="large">i</span></p>
|
||
</div>
|
||
<p>OK, the answer still involves <span class="large">i</span>, but it gives a sensible and <b>consistent</b> answer.</p>
|
||
<p>And <span class="large">i</span> has this interesting property that if we square it (<span class="large">i</span>×<span class="large">i</span>) we get <b>−1</b> which is back to being a Real Number. In fact that is the correct definition:</p>
|
||
<div class="def">
|
||
<p><a href="imaginary-numbers.html">Imaginary Number</a>: A number whose square is a <b>negative</b> Real Number.</p>
|
||
</div>
|
||
<p>And <span class="large">i</span> (the square root of −1) times any Real Number is an Imaginary Number. So these are all Imaginary Numbers:</p>
|
||
<ul>
|
||
<li>3<span class="large">i</span></li>
|
||
<li>−6<span class="large">i</span></li>
|
||
<li>0.05<span class="large">i</span></li>
|
||
<li><span class="times">π</span><span class="large">i</span></li>
|
||
</ul>
|
||
<p>There are also many applications for Imaginary Numbers, for example in the fields of electricity and electronics.</p>
|
||
<h2>Real vs Imaginary Numbers</h2>
|
||
<p>Imaginary Numbers were originally laughed at, and so got the name "imaginary". And Real Numbers got their name to distinguish them from the Imaginary Numbers.</p>
|
||
<p>So the names are just a historical thing. Real Numbers aren't "in the Real World" (in fact, try to find exactly half of something in the real world!) and Imaginary Numbers aren't "just in the Imagination" ... they are both valid and useful types of Numbers!</p>
|
||
<p>In fact they are often used together ...</p>
|
||
<p class="center larger">"what if we put a <b>Real Number</b> and an <b>Imaginary Number</b> together?"</p>
|
||
<h2>Complex Numbers</h2>
|
||
<p>Yes, if we put a Real Number and an Imaginary Number together we get a new type of number called a <a href="complex-numbers.html">Complex Number</a> and here are some examples:</p>
|
||
<ul>
|
||
<li>3 + 2<span class="large">i</span></li>
|
||
<li>27.2 − 11.05<span class="large">i</span></li>
|
||
</ul>
|
||
<p>A Complex Number has a Real Part and an Imaginary Part, but either one could be zero</p>
|
||
<div class="center80">
|
||
<p>So a Real Number is also a Complex Number (with an imaginary part of 0):</p>
|
||
<ul>
|
||
<li>4 is a Complex Number (because it is 4 + 0<span class="large">i</span>)</li>
|
||
</ul>
|
||
<p>and likewise an Imaginary Number is also a Complex Number (with a real part of 0):</p>
|
||
<ul>
|
||
<li>7<span class="large">i</span> is a Complex Number (because it is 0 + 7<span class="large">i</span>)</li>
|
||
</ul>
|
||
</div>
|
||
<p>So the Complex Numbers include all Real Numbers and all Imaginary Numbers, and all combinations of them.</p>
|
||
<p> </p>
|
||
<p class="center large">And that's it!</p>
|
||
<p class="center large">That's all of the most important number types in mathematics.</p>
|
||
<p class="center">From the Counting Numbers through to the Complex Numbers.</p>
|
||
<p><i>There are other types of numbers, because mathematics is a broad subject, but that should do you for now.</i></p>
|
||
<h2>Summary</h2>
|
||
<p>Here they are again:</p>
|
||
<div class="beach">
|
||
<table align="center" width="90%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<th>Type of Number</th>
|
||
<th>Quick Description</th>
|
||
</tr>
|
||
<tr>
|
||
<td>Counting Numbers</td>
|
||
<td>{1, 2, 3, ...}</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Whole Numbers</td>
|
||
<td>{0, 1, 2, 3, ...}</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Integers</td>
|
||
<td>{..., −3, −2, −1, 0, 1, 2, 3, ...}</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Rational Numbers</td>
|
||
<td>p/q : p and q are integers, q is not zero</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Irrational Numbers</td>
|
||
<td>Not Rational</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Real Numbers</td>
|
||
<td>Rationals and Irrationals</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Imaginary Numbers</td>
|
||
<td>Squaring them gives a negative Real Number</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Complex Numbers</td>
|
||
<td>Combinations of Real and Imaginary Numbers</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p> </p>
|
||
<h2>End Notes</h2>
|
||
<h3>History</h3>
|
||
<p>The history of mathematics is very broad, with different cultures (Greeks, Romans, Arabic, Chinese, Indians and European) following different paths, and many claims for <b>"we thought of it first!"</b>, but the general order of discovery I discussed here gives a good idea of it.</p>
|
||
<h3>Questions</h3>
|
||
<p>And isn't it amazing how many times that asking a question, like</p>
|
||
<ul>
|
||
<li><i>"what happens if we count backwards through zero"</i>, or</li>
|
||
<li><i>"what is the exact distance across the diagonal of the square"</i></li>
|
||
</ul>
|
||
<p>first led to disagreement (and even ridicule!), but eventually to amazing breakthroughs in understanding.</p>
|
||
<p><b>I wonder what interesting questions are being asked now?</b></p>
|
||
<h2>Over to You!</h2>
|
||
<p>Here are two questions you can ask when you learn something new:</p>
|
||
<div class="indent50px">
|
||
<h3>Can it go the other way?</h3>
|
||
<ul>
|
||
<li>Positive numbers lead to negative numbers</li>
|
||
<li>Squares lead to square roots</li>
|
||
<li>etc</li>
|
||
</ul>
|
||
<h3>Can I use this with something else I know?</h3>
|
||
<ul>
|
||
<li>If fractions are numbers, can they be added, subtracted, etc?</li>
|
||
<li>Can I take the square root of a complex number? (can you?)</li>
|
||
<li>etc</li>
|
||
</ul>
|
||
</div>
|
||
<p>And one day <b>your</b> questions may lead to a new discovery!</p>
|
||
<p> </p>
|
||
<div class="questions">426,427,429, 2978, 2979, 2980, 2981, 3973, 3974, 3975</div>
|
||
|
||
<div class="related">
|
||
<a href="../mathematics.html">Welcome to Mathematics!</a>
|
||
<a href="../mathematics-language.html">The Language of Mathematics</a>
|
||
<a href="basic-operations.html">Basic Operations</a>
|
||
<a href="../sets/number-types.html">Common Number Sets</a>
|
||
<a href="index.html">Numbers Index</a>
|
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</div>
|
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