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### Puzzles in geometry and algebra

Puzzles in geometry and algebra
According to a Babylonian tablet housed in Berlin, the diagonal of a 40 by 10 rectangle is equal to 40 + 102/. (2 40). An extremely useful approximation rule, the formula employed here (the square root of a2 + b2 can be approximated by a + b2/2a) occurs frequently in later Greek geometric texts. These two root illustrations show how the Babylonians utilised mathematics to analyse geometry. They also demonstrate that the Babylonians, around a millennium before the Greeks, were familiar with the Pythagorean theorem (the relationship between the hypotenuse and the two legs of a right triangle). If you’re having trouble with division, stop stressing and get on over to their site.
Babylonian tablet puzzles sometimes involve finding the base and height of a rectangle given the product and total. The scribe used this data to calculate the following calculation for the disparity: It can be shown that (b h)2 = (b + h)2 4bh. To a similar extent, we could calculate the sum total if we knew the product and the difference. The following equations allowed us to calculate both sides once we had the sum and difference: It may be shown that 2b = (b + h) + (b h) and 2h = (b + h) (b h). The steps are quite similar to those taken when addressing a quadratic equation with a single variable. In some cases, however, the Babylonian scribes employed the same quadratic method that we do today to find solutions to quadratic equations that involve only a single unknown.
However, there are many inconsistencies when using these Babylonian quadratic procedures as early examples of algebra. The lack of algebraic symbols may indicate that the scribes were aware of the generic nature of their solution techniques, but chose to describe them in terms of individual examples rather than the derivation of general equations and identities. Because of this, they were unable to provide convincing explanations for the method they planned to use to fix the issue. Given that similar algorithmic approaches are increasingly commonplace as a result of the widespread availability of computers, their employment of sequential procedures rather than equations is less likely to distract from an evaluation of their work.
Area of a rectangle is equal to the product of its base (b) and height (h) squared, which is a relationship that was well known to the aforementioned Babylonian scribes (h). It’s more probable than not that the third word you select at random will be unreasonable if you pick the first two at random. It is, nevertheless, feasible to identify instances when all three terms are integers, as in (3, 4, 5). (5, 12, 13). (Also known as Pythagorean triples, in some cases.)
The values in the h column are derived from the b and d values since they do not really show on the tablet but were likely part of a section that is no longer there. The d2/h2 values shown in another column (brackets indicate missing or unreadable numbers) disclose the correct sequencing of the lines: [1 59 0] 15, [1 56 56]. 58 14 50 6 15,…,  23 13 46 40. As a result, the angle produced by the diagonal and the base gradually grows from little more than 45 degrees to slightly less than 60 degrees. This scribe knew the basic process for determining all such number triples, like 2d/h = p/q + q/p and 2b/h = p/q q/p for all p and q. (As was explained before in reference to multiplication tables, the p and q values displayed in the table are regular integers that belong to the ordinary set of reciprocals.) While there may be some disagreement among specialists as to the mechanics of the table’s construction, no one disputes the breadth of understanding it displays.
Mathematics for astronomical purposes
It’s possible that the Babylonian sexagesimal approach can do computations several times more sophisticated than those stated in the problem texts. However, its significance was not completely appreciated until the advent of mathematical astronomy in the Seleucid period. One of astronomy’s original purposes was to foretell future events, such as lunar eclipses and changes in the planets’ orbital periods (conjunctions, oppositions, stationary points, and first and last visibility). A method based on summing the terms of an arithmetic progression that began at these coordinates was developed (in degrees of latitude and longitude, according to the apparent annual motion of the Sun). Next, the scribe compiled the data into a table detailing upcoming shifts for as long as was necessary. (Though mathematical in nature, the outcomes are visible; the tabulated numbers provide a linear “zigzag” approximation to the true sinusoidal oscillation.) Without waiting millennia for adequate evidence to specify parameters, astronomers have been able to make a prediction with the use of computer technologies (such as periods, angular range between maximum and lowest values, and so on).
Within a short window of time, the Greeks were exposed to various parts of this system (maybe a century or less). While Hipparchus (2nd century BCE) favoured the geometric approach of his Greek forefathers, he did utilise the Mesopotamians’ sexagesimal system and obtained parameters from them. The Greeks introduced it to the Middle East, and during the Renaissance and early Modern eras in Europe it was widely used in mathematical astronomy. Time and angle calculations still frequently make use of minutes and seconds.
Some aspects of Old Babylonian mathematics may have been known to the Greeks as early as the fifth century BCE, when Greek geometry was still in its infancy. The comparisons made by the specialists are many. For instance, some of the Babylonian quadratic procedures are analogous to the Greek approach known as “application of area” (for more on Greek mathematics, see the section below) (although in a geometric, not arithmetic, form). Perhaps the Greeks also used the Babylonian method of approximating square roots in their geometric calculations, and there were even subtleties in the technical nomenclature that both cultures used. Although the Greeks played an important role, Western mathematics owes much to the earlier Mesopotamians; yet, the precise chronology and conditions of its emergence remain uncertain due to a lack of unambiguous record.

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