Making use of mathematical reasoning and computations

Making use of mathematical reasoning and computations
The Egyptians, like the Romans who came after them, employed the decimal method to writing numbers by writing the digits one through ten and one hundred and one thousand individually and then repeating the procedure. The number 24 was also represented symbolically in mathematics. Although this symbol was used in more formal hieroglyphic writing, such as that found on stone inscriptions and other literature, the scribes who recorded official records on papyrus opted for a shorter, more convenient script called hieratic writing, in which the abbreviation represented the number 24. Visit Their Site If You Need Assistance with Multiplication.
In such a system, performing arithmetic operations like addition and subtraction entails only counting how many of each symbol type are present in the numerical expressions in question and rewriting the expressions in question accordingly. There is no indication in the surviving manuscripts of any special techniques utilized by the scribes to facilitate this. On the other hand, they multiplied by consecutively doubling numbers.
Items in the first column (8, 2, and 1) totaling 11 are checked off as fulfilled. The needed product, 308, may be written as the sum of the multiples of these values. This results in a sum of 308 (or 224 + 56 + 28).
When dividing 308 by 28, the Egyptians went in the other direction. If we consult the same table as previously, we find that 8 produces the largest multiple of 28 that is less than 308 (because 16’s entry is already 448), thus we accept this answer. The next step is to take the original number and remove 8 from it (resulting in 224), and then repeat the process with the resultant number (84). (308). Even though this is greater than the entry at 2, it is reported as correct since it is less than the item at 4. When subtracting 56 from 84, the resulting residual is 28, which also happens to exactly match the entry at 1, we repeat the process and mark it as accurate. The sum of the options selected yields this fraction: (nine, three, four) Equals (eleven) (eleven). In practise, the divisor is always less than the remainder, hence this statement is true.
Considerations of multiples of one of the components by 10, 20,…, or even higher orders of magnitude may help optimise this procedure for larger numbers (100, 1,000,…). (the calculations for these multiples are straightforward in the Egyptian decimal system.) The product of 28 by 27 may be found by listing the multiples of 28 by 1, 2, 4, 8, 10, and 20. Since 27 is the sum of 1, 4, and 20, this problem may be solved by simply putting the proper multiples together.
When working with fractions, only whole numbers may be used as input (that is, fractions that in modern notation are written with 1 as the numerator). In order to indicate the result of dividing 4, which in modern notation is just 4/7, the scribe wrote 1/2 + 1/14. To find quotients in this form, which is essentially an extension of the conventional technique for dividing integers, one need only look at the entries for 2/3, 1/3, 1/6, etc. and 1/2, 1/4, 1/8, etc. until the corresponding multiples of the divisor sum to the dividend. (It’s important to keep in mind that the scribes counted 2/3 even though it’s not a unit fraction.) You may make it as easy as possible (the Rhind papyrus gives us the value for 2/29 as 1/15 + 1/435), or as complicated as you want (you might get the same 2/29 by adding together 1/16 + 1/232 + 1/464, etc.). Many of the papyrus texts are organised as tables, making it simpler to find the appropriate unit-fraction values.

It is possible to do all the math needed to interpret the papyri using just these four procedures. Using simple division, we obtain 1/2 + 1/10 as the answer to the issue “share 6 loaves among 10 men” (Rhind papyrus, problem 3). One particularly clever series of riddles asks, “a number (aha!) and its 7th combination form 19—what is it?” Specifically, (Rhind Papyrus Problem No.24). Assuming the quantity to be 7, we multiply it by 7 to obtain (16 + 1/2 + 1/8) as 11/7 of 7 is 8, not 19. Though they seem to have no direct link to the Egyptian system, many other mathematical systems (including the Chinese, Hindu, Muslim, and Renaissance European) make use of a similar notion (sometimes labelled the approach of “false position” or “false assumption”).

Using the supplied base and height, solvers of the papyri’s geometry problems must identify the shape’s length, breadth, and height. Finding a rectangle with the following proportions (area = 12) and height = 1/2 + 1/4 base is a more challenging task (Golenishchev papyrus, problem 6). Multiplying the inverted ratio by the area yields 16; taking the square root of this number (4) gives the rectangle width, while multiplying the square roots of 1/2 and 1/4 by 4 yields 3 for the height of the rectangle. Solving this expression is identical to solving the related algebraic equation (x 3/4x = 12), except that no letter is replaced for the unknown. Intriguingly (Rhind papyrus, problem 50), one may get the area of a circle by first eliminating 1/9 of the diameter and then squaring. If the diameter is 9, then the area is 64. The scribe chooses 64/81 as the constant of proportionality, /4, since the area of a circle is proportional to the square of its diameter. To put it another way, with an inaccuracy of just 0.6%, this is an excellent estimate. (At roughly 0.04 percent off, it’s not nearly as exact as the traditional estimate of 31/7, first supplied by Archimedes.) 80 percent of 32 is equal to what? But there is nothing in the papyri to suggest that the scribes knew this rule was imprecise.
The surprising result is a rule for the volume of a truncated pyramid (Golenishchev papyrus, problem 14). The scribe calculates these measurements using the following assumptions: a six-foot height, a four-sided base, and a two-sided top. If he were to take the height and divide it by 3, he would get 28, and if he were to multiply that number by 3, he would get 56 (because 28 is equal to 22 plus 24 plus 44). A = (h/3)(a2 + ab + b2) is the answer if the scribe knew the basics. How the scribes arrived at this formula is unknown, however it is possible that they were acquainted with comparable ideas, such as the one for determining the volume of a pyramid (which is equal to one-third the height multiplied by the area of the base).

Egyptian sacking clothing, or seked, dates back to ancient Egypt.
The Egyptians used mathematically equivalent triangles to determine distances. For instance, a pyramid’s seked is written as the number of horizontal palms that would measure one cubit in height (seven palms). Given a seked of 51/4 and a base of 140 cubits, the height is 931/3 cubits (Rhind papyrus, problem 57). It is said that in the sixth century BCE, the Greek sage Thales of Miletus measured the height of the pyramids by observing the length of the shadows they produced (the report derives from Hieronymus, a disciple of Aristotle in the 4th century BCE). The seked calculations, however, suggest that this story hints at an aspect of Egyptian surveying that goes back at least a thousand years before Thales’ time.