Journey into PI

Knowing $$\pi$$ better

The constant $$\pi$$ can be defined as the ratio between the perimeter and the diameter of a circumference $p=\pi d=2\pi r$

The first use of the symbol $$\pi$$ to represent pi is due to William Jones in 1706, and was later adopted by Euler in 1748, from which it became popular and has become the standard notation for this constant.

One can prove that the number $$\pi$$ is irrational and transcendental.

A number is said to be irrational when it cannot be represented by a fraction of two integers, and transcendent if it is not a zero of any non-zero polynomial function of integer coefficients.

For a long time mathematicians believed that all magnitudes were commensurable, which can be translated into modern language by the statement that all numbers were rational. The Greeks demonstrated that the diagonal of the square is not commensurable with the side of the square, which we can express in current language by saying that $$\sqrt{2}$$ cannot be expressed as a ratio of two integers, or is not rational.

The rebellious characteristics of these numbers earned them the name of irrational numbers.

The rational numbers have either a finite decimal expansion (regular) or an infinite periodic decimal expansion (irregular).

As examples, we have $\frac{18241}{148}=123.25$$\frac{423579618}{122563546875}=0.00\,345\,6$ which correspond to regular rational numbers. And irregular rational numbers $\frac{6763}{1232}=5.489\,448\,051\,948\,051\,948\,051\,948\,051\,948\,...$$\frac{122563546875}{423579618}=289.351\,851\,851\,851\,851\,851\,851\,851\,851 ...$ The first of these fractions represents an irregular rational number whose period is $$L = 480519$$ of length $$6$$, and the following is an irregular rational number whose period is $$L = 518$$ of length $$3$$.

The following example seems to contradict what we have just said: the challenge is to find out what is the period of the rational number represented by the fraction $$\frac{368}{491},$$ $\begin{split} \frac{368}{491}=\\ & \hspace-1.8ex 0.749\,490\,835\,030\,549 \,898\, 167\,006\,109\,979\,633\\ & \, 401\,221\,995\,926\,680\,244\, 399\,185\,336\,048\,879\,\\ & 837\, 491\, 067\,209\, 775\, 967\,413\,441\,955\,193\, 482\, \\ &688\,391\,038\,696\,537 \,678\,207\,739\,307\,535\,641\,\\ & 547\,861\,507\,128\, 309\,572\,301\,425\, 661\, 914\,460\,\\ & 285\,132\,382\, 892\,057\,026\,476\,578\, 411\, 405\,295\,\\ &315\,682\,281\, 059\, 063\,136\, 456\,211\,812\,627\, 291\, \\ &242\,362\,525\,458\,248\, 472\,505\,091\,649\,694\,501 \,\\ & 018\,329\,938\,900\,203\, 665\, 987\,780\, 040\,733\,197\,\\ & 556\, 008\, 146\,639\,511\,201\, 629\, 327\,902\,240\,325\,\\ &865\,580\, 448\,065\,173\,116\,089\, 613\, 034\,623\, 217\,\\ &922\, 606\, 924\,643\,584\, 521\,384\, 928\, 716\,904\,276\,\\ &985\,743\, 380\,855\,397\,148\,676\,171\, 079\,429\,735\,\\ &234\, 215\,885\, 947\,046\,843\,177\,189\,409\, 368\, 635\,\\ & 437\,881\,873\,727\, 087\,576\,374\,745\,417\,515\, 274\,\\ & 949\,083\,503\, 054\,989\, 816\, 700\,610\,997\,963\,340\, \\ &122\, 199\,592\,668\,024\,439\, 918\,533\,604\,887\,983\,\\ &706\, 720\, 977\,596\,741\, 344\,195\, 519\, 348\,268\,839\,\\ &103\,869\, 653\, 767\,820\,773\,930\,753\, 564\,154\,786\,\\ & 150\,712\,830\, 957\,230\,142\,566\, 191\,446\, 028\, 513\,\\ & 238\,289\,205\,702\, 647\, 657\,841\,140\,529\,531\, 568\,\\ & 228\,105\,906\,313\,645\, 621\,181\,262\,729\, 124\,236\,\\ & 252\,545\, 824\, 847\,250\,509\,164\,969 \,450\,101\, 832\,\\ &993\,890\,020\, 366\,598\,778\,004\,073\, 319\, 755\,600\,\\& 814\,663\, 951\, 120\,162\,932\,790\,224\, 032\, 586\,558\,\\ &044\,806\,517\, 311\,608\,961\,303\,462\,321\, 792\,260\,\\ &692\,464\, 358\,452 \,138 \,492\,871\,690\, 427\,698\, 574 \,\\ &338\,085\,539\,714\,867\, 617\,107\,942\,973\,523\,421\,\\ & 588\,594\,704\,684\, 317\,718\, 940\, 936\,863\,543\,788\,\\ &187\, 372\, 708\,757\,637\,474\,541\, 751\,527\,494\,908\,\\ &350\,305 \,498\,981, \ldots \end{split}$

Unlike the rational numbers, irrational numbers have an infinite and non-periodic decimal expansion.

Lambert in 1761 and Legendre in 1794 proved that $$\pi$$ is irrational.

In 1882, Lindemann showed that $$\pi$$ is transcendental, i.e. it can not be expressed as a root of an algebraic equation with rational coefficients. It follows that $$\pi$$ could never be built using a ruler and compass (with a finite number of steps). Only numbers which are not transcendental and are of a very particular type, can be constructed in this way.

A direct consequence of this fact implies that one of the most famous geometric problems of antiquity, the squaring of the circle, is not possible.

Despite its simple definition, the number $$\pi$$ appears in many relationships in mathematics, physics and engineering in fields far from themes involving areas of circles and arc lengths.

Computing the value of $$\pi$$ throughout the ages

Approximations for $$\pi$$