Mathematical equations and geometry are very complex
An ancient Babylonian tablet, currently housed in Berlin, has the solution to the diagonal of a 40 by 10 rectangle: 40 + 102/. (2 40). The formula employed here (that the square root of the sum of a2 + b2 may be deduced as a + b2/2a) often occurs in later Greek geometric texts since it is a beneficial approximation rule. These two instances of roots show that the Babylonians had the mathematical prowess to analyze geometric forms. As a bonus, they demonstrate that the Babylonians understood the Pythagorean theorem, which defines the relationship between the hypotenuse and the two legs of a right triangle well over a thousand years before the Greeks used it. If you’re having trouble with mulitplication, stop fretting and get over to their site for multiplication questions.
Rectangle base and height puzzles based on the product and sum are typical on Babylonian tablets. The scribe utilized this to determine that the disparity was best expressed as (b + h)2 = 4bh. Likewise, if we knew the product and the difference, we could easily calculate the total. The two sides might be calculated using the sum and difference as follows: 2b = (b + h) + (b h) and 2h = (b + h) (b h). This process may be compared to solving a generic quadratic in a single variable. There are, however, instances when the Babylonian scribes employed the same quadratic method that we use today to find solutions to quadratic equations that include just a single unknown.
Many people believe that the quadratic procedures used by the Babylonians are early examples of algebra, although there are major inconsistencies between the two theories. There is clear evidence that the scribes understood the generalizability of their approaches. Still, they nonetheless consistently articulated those approaches in particular cases rather than the derivation of universal equations and identities, indicating a lack of algebraic symbolism. As a result, they could not provide convincing arguments for how they tackled the problem or how they fixed it. However, given the widespread availability of computerized approaches conceptually similar to their own, the fact that they used sequential processes rather than equations is less likely to be held against them during evaluation.
The aforementioned Babylonian scribes understood that for every rectangle with a given base (b), height (h), and diagonal (d), the ratio b2 + h2 = d2 always holds (d). It’s more probable that the third word you choose randomly will be unreasonable if you use any two words you like. All three terms may not always be integers; nonetheless, there are cases when this is the case, including (3, 4, 5). (5, 12, 13). (Such answers are sometimes called Pythagorean triples.)
The numbers in the h column were derived from those in the b and d columns since they do not exist on the tablet but were likely part of a portion that is no longer there. Lines [1 59 0] 15, [1 56 56] to be read in the right order: [1 59 0] 15, [1 56 56] (where brackets represent missing or illegible digits). 58 14 50 6 15,…, [1] 23 13 46 40. Therefore, as the process proceeds, the angle produced between the diagonal and the base grows from a little over 45 degrees to a little under 60 degrees. Since 2d/h = p/q + q/p for all p and q, and 2b/h = p/q q/p for all p and q, it’s safe to assume that the scribe was familiar with the basic process for determining all such number triples. As was previously noted in regard to multiplication tables, the p and q values given in the table are regular integers that belong to the standard set of reciprocals. Experts may disagree on the finer points of the table’s design and intended use, but none can dispute the wealth of data it represents.
Science of the stars using math
Babylonian sexagesimal approach may do computations far more complex than those required by the original problem texts. However, its significance was not completely appreciated until the advent of mathematical astronomy in the Seleucid era. 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). They came up with a method for doing so by summing the terms of an arithmetic progression that matched these coordinates (in degrees of latitude and longitude, according to the apparent annual motion of the Sun). Following data collection, a table was produced that detailed each upcoming position for as long as the scribe considered necessary. (Despite being a mathematical procedure, the outcomes are visible; the tabulated numbers indicate a linear “zigzag” approximation to the true sinusoidal oscillation.) As a result of the availability of computational tools, astronomers were able to make a prediction without having to wait millennia for sufficient evidence to establish parameters (such as periods, angular range between maximum and lowest values, and so on).
The Greeks had a head start in understanding some aspects of this system (maybe a century or less). Hipparchus (second century BCE) preferred the geometric technique of his Greek forefathers, but he learned the sexagesimal method of calculation from the Mesopotamians and applied it in his calculations. The Greeks gave it to the Arab scientists in the Middle Ages, and then it came back to Europe and was used extensively in the field of mathematical astronomy throughout the Renaissance and early Modern eras. Minutes and seconds are still often substituted for decimal degrees and radians while performing geometry.
As early as the fifth century BCE, Old Babylonian mathematics may have affected Greek geometry. Several similarities have been noted by experts. One Greek technique, “application of area” (for more on Greek mathematics, see the next section), is analogous to many Babylonian quadratic procedures (although in a geometric, not arithmetic, form). The Babylonian technique of computing square roots was widely utilised in Greek geometric calculations, and there may have been other linguistic and conceptual similarities between the two cultures as well. It is uncertain when and how the Greeks contributed to the formation of contemporary Western mathematics because of a lack of reliable record. However, the ancient Mesopotamians deserve much of the credit.
