Quantitative Reasoning

# Quantitative Reasoning

Let us look at how Quantitative Reasoning works.

# Quantitative Reasoning

The Egyptians, like the Romans who came after them, utilized a decimal method to represent numbers by writing each digit in turn from one to 10 and then one hundred and one thousand. To provide only one example, the number 24 was represented symbolically in mathematics. Hieroglyphic writing, such as that seen on stone inscriptions and other literature, utilized this symbol for the number 24, but the scribes who maintained official records on papyrus employed a shorter, more convenient script called hieratic writing.  Let us have a look at the steps on how to divide 350 by 950.

In such a system, performing arithmetic operations like as addition and subtraction entails only counting the number of each symbol type in the numerical expressions in question and rewriting the expressions in question to reflect the new symbol count. None of the surviving manuscripts disclose the specific techniques the scribes employed to aid with this. On the other hand, they multiplied using a method based on successive doubles.

The first column is checked off if the sum of the values there (in this case, 8, 2, and 1) is 11. Three hundred and eight is the answer, which can be found by adding up the multiples of these numerals. That's why 224 + 56 + 28 Equals 308.

When the Egyptians needed to divide 308 by 28, they did the calculation backwards. When consulting the same table, 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. Then, we use the same method on the remaining after subtracting 8 (224) from the original number (84). (308). While this is greater than the entry at 2 (56), it is still smaller than the item at 4, hence it is rejected. 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 things that have been selected yields this fraction: Eight plus Two plus One Equals (eleven). (In practise, of course, the divisor is always less than the remainder.)

For larger numbers, one may optimise this procedure by considering multiples of one of the components by 10, 20, or even higher orders of magnitude (100, 1,000,). It's simple to calculate these multiples using Egyptian decimal notation. The result of multiplying 28 by 27 may be found by listing the multiples of 28 by 1, 2, 4, 8, 10, and 20 in order. Since the total of 1, 2, 4, and 20 equals 27, putting the proper multiples together yields the right answer of 27.

To do fractional computations, only whole numbers may be used as input (that is, 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 wrote 1/2 + 1/14. Finding quotients in this form is as simple as looking 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 amount to the dividend, which is simply an extension of the conventional procedure for dividing integers. (It's important to remember that despite 2/3's non-unit status, the scribes nevertheless counted it.) Both simpler and more complex methods are possible; for example, the Rhind papyrus gives the answer for 2/29 as 1/15 + 1/435, although the same result might be achieved by adding together smaller numbers such as 1/16 + 1/232 + 1/464, etc. Many of the documents on papyrus are laid out in tables to make it simpler to get the appropriate values for the unit fractions.

You can do all the math needed to decipher the papyri with only these four simple operations. When dividing by 10, the answer is 1/2 + 1/10, which is exactly what is needed to "share 6 loaves among 10 men" (Rhind papyrus, problem 3). One series of riddles uses a clever method, such as "a number (aha) and its 7th combination form 19—what is it?" (The Twenty-fourth Problem on the Rhind Papyrus). Since 11/7 of 7 is 8, and not 19, we double the assumed amount by 7 to obtain (16 + 1/2 + 1/8) below. There seems to be no direct connection between Egyptian and the many other mathematical systems that employ a similar notion (sometimes called the approach of "false position" or "false assumption"), including the Chinese, Hindu, Muslim, and Renaissance European.

Geometry

For the geometry problems in the papyri, you'll need to use the right math on shapes like rectangles and triangles to figure out their perimeters and areas. Finding a rectangle with the following measurements is a more challenging task: the area is 12, and the height is 1/2 plus 1/4 times the base (Golenishchev papyrus, problem 6). To get the length of the rectangle, multiply the inverted ratio by the area, yielding 16; the width can be calculated by taking the square root of this number (4), yielding 3, and the height can be calculated by multiplying the inverted ratio by the square root of this number (4), yielding 2. This method is identical to solving the related algebraic equation (x 3/4x = 12), with the exception that no letter is replaced for the unknown. Area of a circle may be found in an unusual method (Rhind papyrus, problem 50) by throwing away 19 of the diameter and then squared. If the diameter is 9, for instance, then the area is 64. The scribe arbitrarily settled on the value 64/81 for the constant of proportionality, /4, given that the area of a circle is proportional to the square of the diameter. This is a very good estimate, with an error of of roughly 0.6%. (At 0.04 percent off, it's not nearly as exact as the go-to estimate of 31/7, first supplied by Archimedes.) Nothing in the papyri suggests, however, that the scribes knew this rule was imprecise.

Unexpectedly, this work yields a volume rule for a truncated pyramid (Golenishchev papyrus, problem 14). The scribe calculates these proportions by assuming 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 22 and 24 and 44). Assuming the scribe was familiar with standard notation, we may deduce that A = (h/3)(a2 + ab + b2). It's not known how the scribes derived this formula, although they almost certainly knew of comparable ideas, like 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).

The seked was a kind of ancient Egyptian clothing.

"Egyptian taken over"

The Egyptians used conceptually equivalent triangles to measure great distances. When describing the seked of a pyramid, for instance, the number of horizontal palms that would equal a cubit in height is used (seven palms). A base of 140 cubits and a seked of 51/4 yields a height of 931/3 cubits (Rhind papyrus, problem 57). It is said that in the sixth century BCE, the Greek sage Thales of Miletus utilised the length of the shadows produced by pyramids to accurately measure their height (the report derives from Hieronymus, a disciple of Aristotle in the 4th century BCE). Nevertheless, taking into consideration the seked estimates, this story must indicate a component of Egyptian surveying that goes back at least a thousand years before the time of Thales.