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<title>Is It Irrational?</title>
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<h1 align="center">Is It Irrational?</h1>
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<p align="center">Here we look at whether a square root is irrational ... or not!</p>
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<h3>Rational Numbers</h3>
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<p>A "Rational" Number can be written as a "Ratio", or fraction.</p>
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<div class="example">
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<p>Example: <b>1.5</b> is rational, because it can be written as the ratio <b>3/2</b></p>
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</div>
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<div class="example">
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<p>Example: <b>7</b> is rational, because it can be written as the ratio <b>7/1</b></p>
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</div>
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<div class="example">
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<p>Example <b>0.317</b> is rational, because it can be written as the ratio <b>317/1000</b></p>
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</div>
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<p>But some numbers <b>cannot</b> be written as a ratio! </p>
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<p>They are called <b><a href="../irrational-numbers.html">irrational</a></b> (meaning "not rational" instead of "crazy!")</p>
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<h2> The Square Root of 2</h2>
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<p>The square root of 2 is <b>irrational</b>. How do I know? Let me explain ...</p>
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<p> </p>
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<h3>Squaring a Rational Number</h3>
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<p>First, let us see what happens when we <b>square</b> a rational number:</p>
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<p>If the rational number is a/b, then that becomes a<sup>2</sup>/b<sup>2</sup> when squared.</p>
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<div class="example">
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<h3>Example: </h3>
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<p class="center large">(<span class="intbl"><em>3</em><strong>4</strong></span>)<sup>2</sup> = <span class="intbl"><em>3<sup>2</sup></em><strong>4<sup>2</sup></strong></span></p>
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</div>
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<p>Notice that the <a href="../exponent.html">exponent</a> is <b>2</b>, which is an <b>even number</b>.</p>
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<p>But to do this properly we should really break the numbers down into their <a href="../prime-factorization.html">prime factors</a> (any whole number above 1 is prime or can be made by multiplying prime numbers):</p>
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<div class="example">
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<h3>Example: </h3>
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<p class="center large">(<span class="intbl"><em>3</em><strong>4</strong></span>)<sup>2</sup> = (<span class="intbl"><em>3</em><strong>2×2</strong></span>)<sup>2</sup> = <span class="intbl"><em>3<sup>2</sup></em><strong>2<sup>4</sup></strong></span></p>
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</div>
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<p>Notice that the exponents are still even numbers. The 3 has an exponent of 2 (3<sup>2</sup>) and the 2 has an exponent of 4 (2<sup>4</sup>).</p>
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<p>In some cases we may need to simplify the fraction: </p>
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<div class="example">
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<h3>Example: (<span class="intbl"><em>16</em><strong>90</strong></span>)<sup>2</sup></h3>
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<p>Firstly: <b>16</b> = 2×2×2×2 = 2<sup>4</sup>, and <b>90</b> = 2×3×3×5 = 2×3<sup>2</sup>×5</p>
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<p class="center large">(<span class="intbl"><em>16</em><strong>90</strong></span>)<sup>2</sup> = (<span class="intbl"><em>2<sup>4</sup></em><strong>2×3<sup>2</sup>×5</strong></span>)<sup>2</sup></p>
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<p class="center large">= (<span class="intbl"><em>2<sup>3</sup></em><strong>3<sup>2</sup>×5</strong></span>)<sup>2</sup></p>
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<p class="center large">= <span class="intbl"><em>2<sup>6</sup></em><strong>3<sup>4</sup>×5<sup>2</sup></strong></span> </p>
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</div>
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<p>But one thing becomes obvious: every exponent is an <b>even number</b>!</p>
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<p>So we can see that when we square a rational number, the result is made up of prime numbers whose exponents are all <b>even</b> numbers.</p>
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<div class="def">
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<p>When we square a rational number, each prime factor has an <b>even exponent</b>.</p>
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</div>
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<p> </p>
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<h3>Back to 2</h3>
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<p>Now, let us look at the number 2: could this have come about by squaring a rational number?</p>
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<div class="example">
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<h3>As a fraction, 2 is <b>2/1</b></h3>
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<p align="center">Which is <b>2<sup>1</sup>/1<sup>1</sup></b> ,and that has <b>odd exponents</b>!</p>
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<p>Can we get rid of odd exponents?</p>
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<p>We could write 1 as 1<sup>2</sup> (so it has an even exponent), and then we have: </p>
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<p align="center">2 = <b>2<sup>1</sup>/1<sup>2</sup></b></p>
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<p>But there is still an odd exponent (on the 2).</p>
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<p>We can simplify the whole thing to <b>2<sup>1</sup></b>, but still an odd exponent.</p>
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<p>We could even try things like 2 = 4/2 = <b>2<sup>2</sup>/2<sup>1</sup></b>, but we still cannot get rid of an odd exponent</p>
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<p align="center">Oh no, there is always an <b>odd</b> exponent. </p>
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<p> </p>
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<p class="larger">So it could <b>not</b> have been made by squaring a rational number!</p>
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</div>
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<p>This means that the value that was squared to make 2 (ie <b>the square root of 2</b>) cannot be a rational number.</p>
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<p>In other words, the square root of 2 is <b>irrational</b>.</p>
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<p> </p>
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<h3>Try Some More Numbers</h3>
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<div class="example">
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<h3>How about 3? </h3>
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<p>3 is 3/1 = 3<b><sup>1</sup></b></p>
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<p>But the 3 has an exponent of 1, so 3 could not have been made by squaring a rational number, either.</p>
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<p>The square root of 3 is <b>irrational</b></p>
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</div>
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<div class="example">
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<h3>How about 4? </h3>
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<p>4 is 4/1 = 2<b><sup>2</sup></b></p>
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<p><b>Yes!</b> The exponent is an even number! So 4 can be made by squaring a rational number.</p>
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<p>The square root of 4 is <b>rational</b></p>
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</div>
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<p> This idea can also be extended to cube roots, etc.</p>
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<h2>Conclusion</h2>
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<p>To find if the square root of a number is irrational or not, check to see if its prime factors all have <b>even exponents</b>.</p>
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<p>It also shows us there <b>must be</b> irrational numbers (such as the square root of two) ... in case we ever doubted it!</p>
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<p> </p>
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<div class="related"><a href="../irrational-numbers.html">Irrational Numbers</a> <a href="../prime-factorization.html">Prime Factors</a></div>
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