Egyptian Mathematics The Representation of Whole Numbers ----------------------------------- The Egyptians used a numbering system based on 10. There were three different systems used, for hieroglyphics, hieratic or demotic documents. The hieroglyphic format is commonly known and easily described. The Egyptians had a special symbol for 1 and the successive powers of 10: 1: a vertical stroke 10: an upside down "U" 100: a hook or spiral that looked like a "9" 1,000: which we will take to be "X" 10,000 100,000 1,000,000 To represent any whole number, then, you simply used enough of each symbol. For instance, 437 would be 9999 UUU IIIIIII Actually, the Egyptians usually wrote from right to left, and would often "pile up" the digits of a number into two rows, so that it didn't take up so much space. So if you look at a real hieroglypic inscription, the number 437 might actually show up something like this: IIII UU 99 III U 99 Addition and Subtraction ------------------------ To add two numbers in the Egyptian system involved two simple steps, combining and carrying. First, combine all the corresponding digits from both numbers: Thus, if we were adding 437 and 725, 437: 9999 UUU IIIIIII 725: 9999999 UU IIIII ------- ----- ------------ Sum: 99999999999 UUUUU IIIIIIIIIIII The second step was, working from the smallest digits to the largest, to take groups of 10 and "promote" them to one copy of the next digit. This is the same as "carrying". Sum: 99999999999 UUUUU IIIIIIIIIIII Combine extra 1's into 10's ---------- Sum: 99999999999 UUUUU+U II Combine extra 10's into 100's Sum: 99999999999 UUUUUU II Combine extra 100's into 1000's ---------- Sum: X 9 UUUUUU II In other words, the answer is 1,162. Multiplication of Whole Numbers ------------------------------- When we multiply 11 times 35, what we WANT is the same as adding up 35 eleven times, or adding up 11 thirty-five times. So we can always solve such a problem by repeated addition. But one advantage of multiplication is that we can get results much more quickly, especially when the problem is something like 499 time 834, or worse. How we GET the result involves a combination of using a simple multiplication table for single digit products, extending this to multiplication of a multidigit number by a single digit number, using "carries", and extending this to the product of two multidigit numbers, using shifted summation. The Egyptians tried to get the result with no multiplication tables beyond multiplication by 2. So, how could you multiply 11 times 35 this way, and still be more efficient than just adding up 35 eleven times? The Egyptians might have noticed that 11 groups of 35 can be thought of as 1 group of 35, 1 group of 2 * 35, no groups of 4 * 35, and 1 group of 8 * 35. What's helpful about this is that I can easily compute the sizes of these groups by repeated doubling: Problem Grouped Doubling Sum ------- ---------- -------- --- 11 * 35 = 1 * 1 * 35 = 1 * 35 = 35 + 1 * 2 * 35 = 1 * 70 = 70 + 0 * 4 * 35 = 0 * 140 = + 1 * 8 * 35 = 1 * 280 = 280 ----- -- --- 11 * 35 = 11 * 35 = 385 Square Roots ------------ The Egyptians had a way of estimating the square root of a number. This was based on the idea that if a number is close to a perfect square, then there's a good approximation for its square root: If X = A*A + B, then square root of X is approximately A + B / ( 2 * A ) (and similarly if X = A*A - B, square root is approximately A - B / ( 2 * A ) ). For instance, to approximate the square root of 10, we note that 10 is close to 9, a perfect square. So 10 = 9 + 1 = 3*3 + 1 and so the square root is approximately 3 + 1 / 6. The Representation of Fractions ------------------------------- The Egyptians had a peculiar method of representing fractions. In general, a fraction was represented as the sum of unary fractions with distinct denominators. That is, a typical fraction might be 1/2 + 1/7 + 1/14 Note that this is the representation for the fraction we would write as 7/14 + 2/14 + 1/14 = 10/14 = 5/7 However, 5/7 is not a legal Egyptian fraction, since its numerator is not 1. Moreover, we would not be allowed to represent 5/7 as the sum of 5 copies of 1/7: 5/7 = 1/7 + 1/7 + 1/7 + 1/7 + 1/7 because the Egyptian fraction representation required not only that the numerators all be 1's, but that the denominators be DISTINCT! There were a few exceptions to the rule. The Egyptians had special symbols for the fractions 1/2, 2/3 and 3/4. So 2/3 and 3/4 "violated" the rules given above, but in general, fractions were written using the unary numerator and distinct denominator form. In writing, a fraction such as 1/3 was represented simpy by writing the symbol for 3, and then placing a symbol above it, shaped like a lens. Surprisingly, there are an infinite number of ways to take any ordinary fraction A/B and represent it in the Egyptian fraction format. It is an interesting exercise to search for such representations that have the fewest number of terms, or the smallest maximum denominator. Horus Eye Fractions ------------------- An alternative notation was available for representing binary fractions 1/2, 1/4, 1/8, 1/16, 1/32 and 1/64. These fractions were formed, somewhat bizarrely, using individual strokes from an iconic representation known as the Eye of Horus. The Length of a Diagonal of a Square ------------------------------------ The Egyptians were interested in the problem of doubling the area of a square. They realized that if they simply doubled both sides, the area increased by a factor of 4. They reasoned that the correct factor must be between 1 (which left the area unchanged) and 2 (which doubled the area). Trying 1.5 for the factor is an improvement, but it actually causes the area to increase by a factor of 2.25, a little too much. Areas: ----- The Egyptians knew how to compute the area of a square or rectangle. In the example problems that they discuss, they work out the area of a field or of a piece of cloth. It seems as though they also had a formula for the area of an irregular four-sided region. It's not clear, since, at least in the Rhind Papyrus, they are simply doing calculations, without explaining what is going on. But the formula says to take the average of the east and west sides, and multiply that by the average of the north and south sides. Interestingly, this formula is exactly correct for a square, a rectangle, or a figure in which a pair of sides are parallel, but it is not correct in general. In fact, it's easy to make a figure whose sides are all 10 feet long, but which has ANY area between 100 and 0. (Just think of how, if you take out the top and bottom sides, you can flatten a cardboard box.) Triangles: ---------- The story is commonly told that the Egyptians needed to survey their land after the annual Nile flooding. To do so required finding 90 degree angles. One way to do this is to create a right triangle, and this, in turn, is easily done by forming a triangle whose sides are of lengths 3, 4 and 5. Such a triangle is sometimes called an "Egyptian triangle". Right triangles, of course, suggest the area question in two ways. First of all, since a right triangle is half of a rectangle, we know that the area of the Egyptian triangle is 6 square units. However, the computation of the area of a general triangle is somewhat more difficult. By the Pythagorean theorem, we are also directed to consider the fact that 3*3 + 4*4 = 5*5, that is, the areas of squares with sides 3 and 4 will sum to the area of a square of side 5. The Area of a Circle -------------------- The Volume of a Cylinder ------------------------ The Volume of a Pyramid ----------------------- Problem number 79 in the Rhind papyrus -------------------------------------- There is an old English riddle poem that goes As I was going to St Ives, I met a man with seven wives. Each wife had seven sacks, Each sack had seven cats, Each cat had seven kits. Kits, cats, sacks and wives, How many were going to St Ives? But this same problem was written down in the Rhind papyrus, thousands of years ago! The papyrus says, if there are seven houses, and in each house there are seven cats, and each cat eats seven mice, and each mouse would have eaten seven ears of grain, and each ear of grain will grow seven bushels of grain, then how many houses, cats, mice, ears and bushels all together? The Calendar ------------ The Egyptians recorded important historical dates by the name of the Pharoah, and the number of years he had been ruling. Thus, blank occurred in the third year of the reign of ... Because Egyptologists have been able to work out a list of the Pharoahs and the approximate lengths of their reigns going back ... years, this means that many records can be understood very accurately. One problem with recording dates this way is that it is very difficult to compute the length of time between today and an event that happened on a certain day of a certain year of the reign of a certain Pharoah. The only way to do it is to add up the reigns of all the Pharoahs in between, and be extra careful at the ends. The Egyptians devised a very simple "civil" calendar, which made it possible to assign a date to every event, and easily determine the number of days that had elapsed. The Egyptian civil calendar is exactly 365 days long. It is made up of 12 months of 30 days, followed by a 13th month of just 5 days. The months had names, but for now let's just call them "the first month" and so on. Thus, if a pyramid was started on the 3rd day of the 4th month of the 137th year, and finished on the 12th day of the 8th month of the 153rd year, then the number of days involved in building the pyramid can be easily determined. However, because the 5 "funny" days are at the end of the year, the following formula always works: (153-1) * 365 + (8-1) * 30 + 12 - (137-1) * 365 + (3-1) * 30 + 3 Now the civil calendar was slightly out of synch with reality, because the solar year is about a quarter day longer than 365 days. That means that 4 solar years later, the Egyptian civil calendar would not read "January 1" but actually "January 2", and after four more solar years, the Egyptian calendar would be ahead another day. After 1460 solar years, or 1461 Egyptian civil years, the calendars would be in agreement again. The Egyptians knew about this discrepancy, but it didn't matter. They had a separate calendar for keeping track of agriculture and astronomy, which more closely followed the cycle of the stars, the seasons, the weather, and the Nile. The civil calendar was not suitable for agriculture. The yearly flooding of the Nile determined everything that happened in farming. In the civil calendar, the average date of this flooding would gradually drift through the calendar. A different calendar was set up, which tried to follow the sun and moon more closely, so that full moons occurred roughly once a month, and so that the Nile flooding, and all the other tasks of farming, happened at about the same time in the calendar. The first versions of the Egyptian lunar calendar were too simple to do this task, but after some revisions, it was successfully used for agriculture and other seasonal activities. The Egyptian lunar calendar was much more complicated than the civil calendar. Months could have 29 or 30 days, years could have 12 or 13 months, and there was an extra leap day added to the end of certain years. All this was done to keep the lunar calendar roughly aligned with the seasons. (Seasons, Dekans, rising of Sirius, Nilometer, the 36 Dekans, 12 in a night, hours, etc ).