Arithmetic and numerical work

# Arithmetic and numerical work

Like the Romans who came later, the Egyptians employed a decimal method to writing numbers by writing the digits one through ten and one hundred and one thousand individually and then repeating the procedure. A symbolic representation of the number 24 might be found in mathematics. While this symbol was utilized in more formal hieroglyphic writing, such as that seen on stone inscriptions and other literature, the scribes who maintained official records on papyrus used a shorter, more convenient script called hieratic writing, in which the abbreviation denoted the number 24. Their website is the place to go if you're struggling with multiplication. Explain the divisibility of 30 by 13.

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 words in the query accordingly. The surviving manuscripts do not show what special techniques the scribes may have utilized to aid with this. Instead, they relied on a method of multiplication based on sequential doublings.

Items in the first column (8, 2, and 1) adding up to 11 are satisfied. The desired product, in this instance 308, may be calculated by adding up the multiples of these values. In this case, 224 plus 56 plus 28 is equal to 308.

To divide 308 by 28, the Egyptians did the calculation backward. Using the same method, we find that 8 produces the most significant multiple of 28, less than 308 (because 16's entry is already 448). Thus, we accept this answer. Then, we repeat Steps 1 through 3 with the remaining after removing 8 (224) from the initial number (84). (308). Though this is greater than the entry at 2 (56), it is reported as incorrect since it is smaller than the item at 4. The technique is repeated for the remainder obtained by subtracting 56 from the original rest of 84, which is 28, and also happens to exactly match the entry at 1, which is subsequently marked as correct. Here's the fraction that results from combining all the currently checked things: Addition of 8+2+1 = (eleven). In practice, 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 optimize this procedure for more significant numbers (100, 1,000,...). These multiples are simple to calculate in Egyptian decimal notation. The result of multiplying 28 by 27 may be found by enumerating the multiples of 28 by 1, 2, 4, 8, 10, and 20. Because the total of entries 1, 2, 4, and 20 equals 27, calculating the proper answer is as easy as adding the relevant multiples together.

There is a restriction on using anything except whole numbers in fraction computations (fractions that in modern notation are written with 1 as the numerator). For the result of dividing 4 by 7, which is just 4/7 in modern notation, the scribe instead wrote 1/2 + 1/14. To find quotients in this form, which is merely an extension of the conventional technique for dividing integers, we 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. Although 2/3 is not a unit fraction, it was tallied as such by the scribes. 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 difficult 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 organized in tables to make it simpler to find the relevant unit-fraction values.

You can do all the math needed to decipher the papyri with only these four simple operations. By dividing, we obtain 1/2 + 1/10, the solution to the "share 6 loaves among 10 men" (Rhind papyrus, problem 3) equation. One particularly clever set of riddles reads as follows: "a number (aha) and its 7th combination form 19—what is it?" (The Twenty-fourth Problem Listed on the Rhind Papyrus). Since 11/7 of 7 is 8, and not 19, we can simplify this by assuming the quantity is 7, and multiplying that by 7 to obtain (16 + 1/2 + 1/8) Although there seems to be no direct connection between the Egyptian and other arithmetic systems (including the Chinese, Hindu, Muslim, and Renaissance European), many of these systems utilise a similar notion (sometimes called the approach of "false position" or "false assumption").

Geometry

Finding the dimensions of shapes like rectangles and triangles from their base and height is a common geometry challenge in the papyri, and it requires the use of suitable mathematical processes. Finding a rectangle with the given measurements (area = 12; 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, and bearing the halves of 1/2 and 1/4 by 4 yields 3 for the height. Solving the related algebraic equation (x 3/4x = 12) follows the same steps, without the step in which a letter is replaced for the unknown. An interesting method for determining the circle's area (Rhind papyrus, problem 50) involves throwing away one-ninth of the diameter and then squaring the result. Using the example of a circle with a diameter of 9, the area would be 64. The scribe arbitrarily settled on the value of 64/81 for the constant of proportionality, /4, given that the area of a circle is proportional to the square of the diameter. The error is just 0.6%, making this a very accurate estimate. (At roughly 0.04 percent off, it's not nearly as exact as the traditional estimate of 31/7, first supplied by Archimedes.) If the scribes knew this rule was imprecise, however, they did not indicate it in the papyri.

Specifically, a rule for the volume of a truncated pyramid is found, which is a somewhat surprising result (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. He obtains 28 by dividing the height by 3, and 56 by multiplying that by 3 (where 28 is determined by adding together 22+24+44). Assuming that the scribe was familiar with standard notational practises, we may write A = (h/3)(a2 + ab + b2). The scribes' method of arriving at this formula is unknown, although they very certainly knew of comparable ideas, such as the one used to determine the volume of a pyramid (which is equal to one-third the height multiplied by the area of the base).

Seked, a kind of ancient Egyptian clothing,

We sacked Egypt.

The Egyptians used mathematically equivalent triangles to measure distances. For instance, a pyramid's seked is written as the number of horizontal palms that would measure one cubit if it were at ground level (seven palms). Assuming a seked of 51/4 and a base of 140 cubits, the height is 931/3 cubits (Rhind papyrus, problem 57). According to myth, the great Greek sage Thales of Miletus studied the length of the shadows produced by pyramids to calculate their precise height in the sixth century BCE (the report derives from Hieronymus, a disciple of Aristotle in the 4th century BCE). However, considering the seked estimates, this narrative must indicate a component of Egyptian surveying that goes back at least a thousand years before the time of Thales.