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      <text>9.4.2 (History of the binomial theorem)</text>
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      <text><![CDATA[<h3>History of the Binomial Theorem</h3>
<p>Since the Binomial Theorem at its core, is concerned with expanding brackets into polynomials, aspects of its theory were known into antiquity as polynomials such as cubics and quadratics were known to the ancient Egyptians, Babylonians and Greeks. However, the combinatorial implications of the Binomial theorem were not fully explored until much later in 17th and 18th century Europe.</p>
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<p>One of the earliest results related to the Binomial theorem was stated by the Greek geometer Euclid in the 4th century BCE. Specifically he mentioned the Binomial expansion for the case where the power is \(2\), that is he gave the expansion of the brackets \((x+y)^2\) in the context of the study of quadratics rather than the context of combinatorics, which would not be discovered until much later. As mentioned in previous weeks, the ancient Greek mathematicians were mainly geometers, and mistrusted (or flatly refused to accept) concepts that could not be translated into geometry. The Greek study of Binomials was somewhat limited by this view. The figure below demonstrates the geometric interpretation of the Binomial Theorem.</p>
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<p><br /> <img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Geometric%20binomial%20theorem.png" alt="diagram showing the geometric interpretation of the binomial theorem" width="438" height="328" /></p>
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<p>The case of the Binomial theorem for cubes can be traced back to Indian mathematicians in around the 6th century CE. The idea of Binomial coefficients representing the combinatorial quantities of 'choosing r objects' out of a total of n objects also appears to have originated in India. The earliest know reference to this problem is by Pingala in 200BCE who seemed to give a method for its solution. This method was eventually explained in the 10th century CE by Halayudha, using a model which we now call Pascal's triangle. Evidence suggests, however that by the 6th century CE, Indian mathematicians knew how to express binomial coefficients as the factorials \(\frac{n!}{(n-k)!k!}\), which is interesting as it appears that Binomial coefficients may have been discovered prior to Pascal's triangle. However, whilst we are not certain of this, we do know that a statement of the rule for binomial coefficients was explicitly stated in the 12th century by Bhaskara.</p>
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<p>The next major advances then came with the medieval Arabic empire. The crowning achievement of their studies into binomials was the first full formulation of the Binomial Theorem and the table of Binomial coefficients by Al-Karaji. Al-karaji also described the triangular nature of the coefficients and provided a proof of the the Binomial Theorem and Pascal's triangle. Later, in the 16th century the German monk Michael Stifel was the first person to coin the term 'Binomial coefficient', he then showed how to use these coefficients to expand brackets of the form \((a+b)^n\) in terms of entries from Pascal's triangle. Blaise Pascal actually studied the triangle in depth in around 1665 in his treatise <em>Traité du triangle arithmétique</em> which is why the eponymous triangle bears his name. Actually the structure of the triangle its self was known to renaissance mathematicians earlier. It is Issac Newton who is credited for the generalisation of the Binomial Theorem to negative and rational powers.</p>
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<p>The next major advance in the study of the Binomial Theorem, was its application to the fields of probability and statistics. This began in the early 18th century with the findings of the Swiss mathematician Jacob Bernoulli, who was the firts European to discover that the probability of k outcomes from n repetitions is given by the kth coefficient in the Binomial expansion of \((p+q)^n\), where \(q = 1-p\). This discovery led to the development of the <em>Binomial Distribution</em> which we will discuss in the next section. This is now a standard tool in the armoury of a statistician as it describes a number of discrete probabilities.</p>]]></text>
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