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diff --git a/site/posts/17-fractions-and-simplification-explained.md b/site/posts/17-fractions-and-simplification-explained.md new file mode 100644 index 0000000..e59bd81 --- /dev/null +++ b/site/posts/17-fractions-and-simplification-explained.md @@ -0,0 +1,415 @@ + + title: "Fractions and simplification, explained" + permalink: '17-fractions-and-simplification-explained' + date: {m: 7, d: 1, y: 2017} + categories: + - 'text' + +--- + +# Fractions and simplification, explained + +I've always found fractions to be a little bit magical; things like "doing the +same multiplication to the numerator and denominator gets you a fraction of +equal value" and the process of simplifying fractions have been useful, but it +hasn't been very clear *how* they work. So in this I set out on the silly, +little, but fun quest to do define the way fractions work using plain old +math.. + +--- + +A *fraction* is sort of like a number made of two parts. For example, we might +have a fraction made of the numbers 1 and 2, and we would write that as +<code class='math' id='math-frac-1_2'>\frac{1}{2}</code>. We could have another +fraction made of the numbers 7 and 9, and that would be written as +<code class='math' id='math-frac-7_9'>\frac{7}{9}</code>. + +The *value* of a fraction is simply the fraction's top number (we call this its +*numerator*) divided by its bottom number (the *denominator*): +<code class='math' id='math-fracval-9_3'>\frac{9}{3} = 9 \div 3 = 3</code>, +<code class='math' id='math-fracval-100_20'>\frac{100}{20} = 100 \div 20 = 5 +</code>, +and so on. (To *evaluate* a fraction just means to replace that fraction with +its value.) + +--- + +You can do lots of cool things with fractions, but we're interested in +*multiplying* them. + +Hopefully you already know how to multiply two normal numbers - for example, +you should already know that +<code class='math' id='math-mul-3x7'>3 \times 7 = 21</code>, +<code class='math' id='math-mul-2x4'>2 \times 4 = 8</code>, and so on. + +Multiplying two fractions is actually pretty simple - we just need to multiply +the two numerators and the two denominators together, and we get another +fraction. For example, to multiply +<code class='math' id='math-multiply-two-fractions-statement'> + \frac{1}{2} \times \frac{4}{3} +</code>: + +<pre class='math' id='math-multiply-two-fractions-solution'> + \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} +</pre> + +We can also multiply other fractions: + +<pre class='math' id='math-multiply-two-fractions-2'> + \frac{7}{5} \times \frac{2}{100} = \frac{7 \times 2}{5 \times 100} = \frac{14}{500} +</pre> + +<pre class='math' id='math-multiply-two-fractions-3'> + \frac{4}{6} \times \frac{1}{3} = \frac{4 \times 1}{6 \times 3} = \frac{4}{18} +</pre> + +..and so on. + +(We've already said that fractions can be made out of two numbers. But we made +the fraction +<code class='math' id='math-frac-1x4-2x3'>\frac{1 \times 4}{2 \times 3}</code> +above - how did that work? Well, it makes sense if you think of +<code class='math' id='math-expr-2x3'>2 \times 3</code> and +<code class='math' id='math-expr-1x4'>1 \times 4</code> +as numbers. In fact you don't need to worry about what those values are (even +though we know them to be 6 and 4); just that they can be used in place of +numbers. If you would like to get very technical, we could say that a fraction +is made of two *expressions*; and an expression is just a number, like 7, or a +calculation, like <code class='math' id='math-expr-6x5'>6 \times 5</code> or +<code class='math' id='math-expr-3+7'>3 + 7</code>.) + +--- + +What if we want to multiply a fraction and a normal whole number? + +<pre class='math' id='math-frac-1-2-x4'> + \frac{1}{2} \times 4 +</pre> + +The trick is that you need to turn the whole number into a fraction, somehow. +The fraction we are creating must have an *equal value* to our whole number, +or else we cannot use it in place of that number. + +It is true that for any number, dividing that number by 1 does not change it: +<code class='math' id='math-5div1-eq-5'>5 \div 1 = 5</code>, +<code class='math' id='math-4div1-eq-4'>4 \div 1 = 4</code>, etc. Since we +already know that the value of a fraction is gotten by dividing the numerator +by the denominator, we can create a fraction from any division problem: simply +use the first number (5, 4, etc.) as the numerator and the second number (1) +as the denominator. +<code class='math' id='math-5-eq-5div1'>5 = \frac{5}{1}</code>, +<code class='math' id='math-4-eq-4div1'>4 = \frac{4}{1}</code>, and so on. + +Now we know that any number <code class='math' id='math-x4'>\times 4</code> +is equal to that same number +<code class='math' id='math-x-frac-4-1'>\times \frac{4}{1}</code>, so we may +put our <code class='math' id='math-frac-4-1'>\frac{4}{1}</code> fraction +into our calculation: + +<pre class='math' id='math-using-frac-4-1'> + \frac{1}{2} \times 4 = \frac{1}{2} \times \frac{4}{1} +</pre> + +And we already know how to multiply two fractions: + +<pre class='math' id='math-multiplying-frac-4-1'> + \frac{1}{2} \times \frac{4}{1} = \frac{1 \times 4}{2 \times 1} = \frac{4}{2} +</pre> + +--- + +We're almost ready to try out something interesting, but first we need to +understand one more concept - that any number multiplied by 1 is the original +number: <code class='math' id='math-4x1-eq-4-again'>4 \times 1 = 4</code>, +<code class='math' id='math-3x1-eq-3-again'>3 \times 1 = 3</code>, +<code class='math' id='math-999x1-eq-999-again'>999 \times 1 = 999</code>. +(We already know this from how multiplying whole numbers always works, of +course.) + +We can apply the same concept to fractions using what we now know about +multiplying a fraction by a whole number: + +<pre class='math' id='math-using-frac-1-1'> + \frac{4}{7} \times 1 = \frac{4}{7} \times \frac{1}{1} = \frac{4 \times 1}{7 \times 1} = \frac{4}{7} +</pre> + +This makes sense because we know that +<code class='math' id='math-1-eq-frac-1-1'>1 = \frac{1}{1}</code> (from our +rule that for any number, we can make a fraction that is equal to that number +by using it as the numerator of the fraction and 1 as the denominator). + +--- + +We can proceed now to define another rule: that for any fraction where the +numerator and denominator are the same, the fraction is equal to 1. For +example, <code class='math' id='math-frac-5-5-eq-1'>\frac{5}{5} = 1</code>, +<code class='math' id='math-frac-1-1-eq-1'>\frac{1}{1} = 1</code>, +<code class='math' id='math-frac-105-105-eq-1'>\frac{105}{105} = 1</code>. + +This makes sense because we know that any number divided by itself equals 1 +(that is, that any number "fits into" itself exactly 1 time). Using our rule +for the value of a fraction, we may write: +<code class='math' id='math-5div5'>\frac{5}{5} = 5 \div 5 = 1</code>, +<code class='math' id='math-1div1'>\frac{1}{1} = 1 \div 1 = 1</code>, and so +on. + +This means that we know how to replace any 1 in our calculations with a +fraction of equal value; let's try that with our "multiplication by 1 equals +itself" rule using a fraction: + +<pre class='math' id='math-double-fraction-parts'> + \begin{align*} + \frac{3}{5} \times 1 &= \frac{3}{5} \\ + \frac{3}{5} \times \frac{2}{2} &= \frac{3}{5} \\ + \frac{3 \times 2}{5 \times 2} &= \frac{3}{5} \\ + \frac{6}{10} &= \frac{3}{5} + \end{align*} +</pre> + +Ah! How peculiar. This reveals that the fraction +<code class='math' id='math-frac-6-10'>\frac{6}{10}</code> is actually equal +to the fraction <code class='math' id='math-frac-3-5'>\frac{3}{5}</code>. + +In fact, we can multiply any fraction and 1, or an equal value, and get a new +fraction that is equivalent to the first fraction: + +<pre class='math' id='math-frac-7-9-times-frac-3-3'> + \begin{align*} + \frac{7}{9} \times 1 &= \frac{7}{9} \\ + \frac{7}{9} \times \frac{3}{3} &= \frac{7}{9} \\ + \frac{7 \times 3}{9 \times 3} &= \frac{7}{9} \\ + \frac{21}{27} &= \frac{7}{9} + \end{align*} +</pre> + +<pre class='math' id='math-frac-4-3-times-frac-5-5'> + \begin{align*} + \frac{4}{3} \times 1 &= \frac{4}{3} \\ + \frac{4}{3} \times \frac{5}{5} &= \frac{4}{3} \\ + \frac{4 \times 5}{3 \times 5} &= \frac{4}{3} \\ + \frac{20}{15} &= \frac{4}{3} + \end{align*} +</pre> + +<pre class='math' id='math-frac-99-150-times-frac-1-1'> + \begin{align*} + \frac{99}{150} \times 1 &= \frac{99}{150} \\ + \frac{99}{150} \times \frac{1}{1} &= \frac{99}{150} \\ + \frac{99 \times 1}{150 \times 1} &= \frac{99}{150} \\ + \frac{99}{150} &= \frac{99}{150} + \end{align*} +</pre> + +<pre class='math' id='math-frac-2-4-times-frac-2-2'> + \begin{align*} + \frac{2}{4} \times 1 &= \frac{2}{4} \\ + \frac{2}{4} \times \frac{2}{2} &= \frac{2}{4} \\ + \frac{2 \times 2}{4 \times 2} &= \frac{2}{4} \\ + \frac{4}{8} &= \frac{2}{4} + \end{align*} +</pre> + +..And so on. + +Now - we have already said that for any fraction where the top and bottom are +equal, that fraction is equal to 1. What if we put two equal fractions inside +of our fraction? + +<pre class='math' id='math-nested-fraction'> + \frac{\frac{3}{9}}{\frac{3}{9}} +</pre> + +This is still equal to 1, because the numerator and denominator of the +"big" fraction are equal. (It *is* true that +<code class='math' id='math-frac-3-9-eq-frac-3-9'>\frac{3}{9} = \frac{3}{9}</code>, +of course!) + +Likewise, fractions such as +<code class='math' id='math-frac-frac-1-2-frac-1-2'>\frac{\frac{1}{2}}{\frac{1}{2}}</code> +and +<code class='math' id='math-frac-frac-5-4-frac-5-4'>\frac{\frac{5}{4}}{\frac{5}{4}}</code> +are also equal to 1. + +--- + +Now we can use all we've learned so far to try this: + +<pre class='math' id='math-frac-8-4-times-halfer'> + \begin{align*} + \frac{8}{4} \times 1 &= \frac{8}{4} \\ + \frac{8}{4} \times \frac{\frac{1}{2}}{\frac{1}{2}} &= \frac{8}{4} + \end{align*} +</pre> + +But how do we multiply those two fractions? Well, we can multiply them exactly +the way we would usually multiply fractions - multiply the two numerators +together and the two denominators together: + +<pre class='math' id='math-mix-frac-8-4-times-halfer'> + \frac{8}{4} \times \frac{\frac{1}{2}}{\frac{1}{2}} = + \frac{8 \times \frac{1}{2}}{4 \times \frac{1}{2}} +</pre> + +We already know how to multiply a whole number and a fraction together - just +convert the whole number to a fraction by putting the whole number on top and 1 +on the bottom: + +<pre class='math' id='math-solve-frac-8-4-times-halfer'> + \frac{8 \times \frac{1}{2}}{4 \times \frac{1}{2}} = + \frac{\frac{8}{1} \times \frac{1}{2}}{\frac{4}{1} \times \frac{1}{2}} = + \frac{\frac{8 \times 1}{1 \times 2}}{\frac{4 \times 1}{1 \times 2}} = + \frac{\frac{8}{2}}{\frac{4}{2}} +</pre> + +We are nearly completed; we may simply insert this fraction back into our +calculation: + +<pre class='math' id='math-frac-substitute-halved-fraction'> + \begin{align*} + \frac{8}{4} \times \frac{\frac{1}{2}}{\frac{1}{2}} &= \frac{8}{4} \\ + \frac{\frac{8}{2}}{\frac{4}{2}} &= \frac{8}{4} + \end{align*} +</pre> + +In order to actually make this useful, we must evaluate the fractions that are +in the top and bottom of our newly-created "big" fraction: + +<pre class='math' id='math-frac-solve-halved-fraction'> + \frac{\frac{8}{2}}{\frac{4}{2}} = \frac{8 \div 2}{4 \div 2} = \frac{4}{2} +</pre> + +And then we may put this back into our main calculation: + +<pre class='math' id='math-simplify-conclusion'> + \begin{align*} + \frac{\frac{8}{2}}{\frac{4}{2}} &= \frac{8}{4} \\ + \frac{4}{2} &= \frac{8}{4} + \end{align*} +</pre> + +As you can see, we have gone from one fraction, in this case +<code class='math' id='math-frac-8-4'>\frac{8}{4}</code>, to a fraction of +equal value but smaller numerators and denominators, +<code class='math' id='math-frac-4-2'>\frac{4}{2}</code>. This is what is +known as *simplifying* a fraction. + +We can show these are of equal value simply by comparing their values: +<code class='math' id='math-frac-8-4-eq-2'>\frac{8}{4} = 8 \div 4 = 2</code>, +and +<code class='math' id='math-frac-4-2-eq-2'>\frac{4}{2} = 4 \div 2 = 2</code>. +We say that because the fractions have an equal value, they are *proportional*. + +--- + +The *greatest common divisor* of two numbers is the greatest whole number you +can divide both numbers by and get two resulting whole numbers. There are +various methods of finding it; we write it with the notation +<code class='math' id='math-gcd-a-b'>\DeclareMathOperator{\gcd}{gcd} \gcd(a, b)</code> +(where <code class='math' id='math-variable-a'>a</code> and +<code class='math' id='math-variable-b'>b</code> are whole number values). For +example, +<code class='math' id='math-gcd-20-15'>\DeclareMathOperator{\gcd}{gcd} \gcd(20, 15) = 5</code> +because dividing <code class='math' id='math-20div5'>20 \div 5 = 4</code> and +<code class='math' id='math-15div5'>15 \div 5 = 3</code> both get us whole +numbers (4 and 3), and there is no greater number that both 20 and 15 can be +divided by to get whole numbers. + +The greatest common factor can be used inside of fraction simplification to +get the "completely" simplified value of any fraction. For example: + +<pre class='math' id='math-gcf-completely-simplified'> + \begin{equation} + \begin{split} + \frac{70}{42} &= \frac{70}{42} \times 1 \\ + &= \frac{70}{42} \times \frac{\frac{1}{\gcd(70,42)}}{\frac{1}{\gcd(70,42)}} \\ + &= \frac{70}{42} \times \frac{\frac{1}{14}}{\frac{1}{14}} \\ + &= \frac{70 \times \frac{1}{14}}{42 \times \frac{1}{14}} \\ + &= \frac{5}{3} + \end{split} + \end{equation} +</pre> + +(I skipped a couple of steps in multiplying the values in the "big" fraction +to keep a focus on the important part, which was applying the greatest common +divisor.) + +Another example: + +<pre class='math' id='math-gcf-simplify-frac-8-4'> + \begin{equation} + \begin{split} + \frac{8}{4} &= \frac{8}{4} \times 1 \\ + &= \frac{8}{4} \times \frac{\frac{1}{\gcd(8,4)}}{\frac{1}{\gcd(8,4)}} \\ + &= \frac{8}{4} \times \frac{\frac{1}{4}}{\frac{1}{4}} \\ + &= \frac{8 \times \frac{1}{4}}{4 \times \frac{1}{4}} \\ + &= \frac{2}{1} \\ + &= 2 + \end{split} + \end{equation} +</pre> + +This time we get the value of +<code class='math' id='math-frac-2-1'>\frac{2}{1}</code>, which is 2, and use +that as our simplified value. + +--- + +All of the above can be written in elegant and general algebra-like math. + +**Value of a fraction:** + +<pre class='math' id='math-value-of-a-fraction'> + \frac{a}{b} = a \div b +</pre> + +**Fractions from values using denominator 1:** + +<pre class='math' id='math-fractions-from-values-using-denominator-1'> + n = n \div 1 = \frac{n}{1} +</pre> + +**Multiply two fractions:** + +<pre class='math' id='math-multiply-two-fractions'> + \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} +</pre> + +**Multiply a fraction and a value:** + +<pre class='math' id='math-multiply-fraction-and-value'> + \frac{a}{b} \times n = + \frac{a}{b} \times \frac{n}{1} = + \frac{a \times n}{b \times 1} = + \frac{a \times n}{b} +</pre> + +**Simplify a fraction (completely):** + +<pre class='math' id='math-completely-simplify-fraction'> + \begin{equation} + \begin{split} + \frac{a}{b} &= \frac{a}{b} \times 1 \\ + &= \frac{a}{b} \times \frac{1 \div \gcd(a,b)}{1 \div \gcd(a,b)} \\ + &= \frac{a \times 1 \div \gcd(a,b)}{b \times 1 \div \gcd(a,b)} \\ + &= \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)} + \end{split} + \end{equation} +</pre> + +Alternate: + +<pre class='math' id='math-completel-simplify-fraction-alt'> + \begin{equation} + \begin{split} + \frac{a}{b} &= \frac{a}{b} \times 1 \\ + &= \left( \frac{a}{b} \times \frac{\frac{1}{\gcd(a,b)}}{\frac{1}{\gcd(a,b)}} \right) \\ + &= \left( \frac{a \times \frac{1}{\gcd(a,b)}}{b \times \frac{1}{\gcd(a,b)}} \right) \\ + &= \left( \frac{\frac{a}{1} \times \frac{1}{\gcd(a,b)}}{\frac{b}{1} \times \frac{1}{\gcd(a,b)}} \right) \\ + &= \left( \frac{\frac{a \times 1}{1 \times \gcd(a,b)}}{\frac{b \times 1}{1 \times \gcd(a,b)}} \right) \\ + &= \left( \frac{\frac{a}{\gcd(a,b)}}{\frac{b}{\gcd(a,b)}} \right) + \end{split} + \end{equation} +</pre> + +(The parentheses around each step are only present to clarify the separate +steps; they don't actually mean anything.) |