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Chapter 1: Problem 1

Represent 375 and 4856 in Egyptian hieroglyphics and Babylonian cuneiform.

Short Answer

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Question: Represent the given numbers (375 and 4856) in Egyptian hieroglyphics and Babylonian cuneiform. Answer: Egyptian hieroglyphics: 375: 5 Vertical Strokes, 7 Horizontal Strokes, 3 Coiled Ropes (bottom to top) 4856: 6 Vertical Strokes, 5 Horizontal Strokes, 8 Coiled Ropes, 4 Lotus Flowers (bottom to top) Babylonian cuneiform: 375: 6 Large Wedges [Space] 1 Large Wedge, 5 Wedge Tips (left to right) 4856: 1 Large Wedge [Space] 2 Large Wedges [Space] 3 Large Wedges, 6 Wedge Tips (left to right)

Step by step solution

01

Representing 375 in Egyptian Hieroglyphics

To represent 375 in Egyptian hieroglyphics, we will break the number down into its units, tens, and hundreds. 375 = 300 + 70 + 5 In Egyptian hieroglyphics: - The number 1 is represented by a single vertical stroke - The number 10 is represented by a horizontal stroke - The number 100 is represented by a coiled rope symbol For 375, we will draw: - Three coiled ropes for the value of 300 - Seven horizontal strokes for the value of 70 - Five vertical strokes for the value of 5 Therefore, 375 in Egyptian hieroglyphics will look like the following arrangement of symbols (bottom to top): ``` 5 Vertical Strokes 7 Horizontal Strokes 3 Coiled Ropes ```
02

Representing 4856 in Egyptian Hieroglyphics

To represent 4856 in Egyptian hieroglyphics, we will break the number down into its thousands, hundreds, tens, and units: 4856 = 4000 + 800 + 50 + 6 In Egyptian hieroglyphics: - The number 1000 is represented by the lotus flower symbol For 4856, we will draw: - Four lotus flower symbols for the value of 4000 - Eight coiled ropes for the value of 800 - Five horizontal strokes for the value of 50 - Six vertical strokes for the value of 6 Therefore, 4856 in Egyptian hieroglyphics will appear like the following arrangement of symbols (bottom to top): ``` 6 Vertical Strokes 5 Horizontal Strokes 8 Coiled Ropes 4 Lotus Flowers ```
03

Representing 375 in Babylonian Cuneiform

To represent 375 in Babylonian cuneiform, we will convert the number into base-60: 375 = (6 x 60^1) + 15 In Babylonian cuneiform: - A single wedge tip will indicate values 1-9 - A larger wedge will indicate values 10-50 For 375, we will draw: - Six large wedges for the value of 6 - A space to separate the base-60 digits - One large wedge and five wedge tips for the value of 15 Therefore, 375 in Babylonian cuneiform will be represented by the following arrangement of symbols (left to right): ``` 6 Large Wedges [Space] 1 Large Wedge, 5 Wedge Tips ```
04

Representing 4856 in Babylonian Cuneiform

To represent 4856 in Babylonian cuneiform, we will convert the number into base-60: 4856 = (1 x 60^2) + (20 x 60^1) + 36 For 4856, we will draw: - One large wedge for the value of 1 - A space to separate the base-60 digits - Two large wedges for the value of 20 - Another space to separate the base-60 digits - Three large wedges and six wedge tips for the value of 36 Therefore, 4856 in Babylonian cuneiform will be represented by the following arrangement of symbols (left to right): ``` 1 Large Wedge [Space] 2 Large Wedges [Space] 3 Large Wedges, 6 Wedge Tips ```

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Egyptian Hieroglyphics
Ancient Egyptians used a fascinating system of writing numbers with symbols called hieroglyphics. Numbers were represented using distinct symbols for different values, allowing people to visually recognize large and small numbers at a glance.

Here鈥檚 how they did it:
  • Units: A vertical stroke (|) denoted the number 1. So, for the number 5, you'd see five vertical strokes.
  • Tens: A horizontal stroke equaled 10. Therefore, the number 70 would be illustrated with seven horizontal strokes.
  • Hundreds: A coiled rope was used for 100. To show the number 300, three coiled ropes would be drawn.
  • Thousands: The lotus flower symbolized 1000. Thus, 4000 required four lotus flowers.
This system was pictorial and relatively straightforward, allowing Egyptians to convey numbers with a combination of these symbols in a clear and logical structure.
Babylonian Cuneiform
The Babylonians introduced an innovative way of writing numbers called cuneiform, which was marked by wedge-shaped impressions in clay. Their system was not just symbolic but also positional, relying on a base-60 system that differed from our usual base-10.

In Babylonian cuneiform:
  • Units 1-9: These were represented with small wedge-shaped marks.
  • Tens 10-50: Larger wedges were used to symbolize numbers in this range.
The clever use of spacing between groups of wedges allowed Babylonians to signify different orders of magnitude, akin to how we use commas to separate thousands in our numeral system.
Base-60 Number System
The Babylonians' number system, known as sexagesimal or base-60, was a groundbreaking method that allowed complex calculations. Unlike our familiar base-10, base-60 uses 60 symbols to represent each order of magnitude.

This system's advantages included ease in dividing by 3, 4, and 6 鈥� common divisors in trade and astronomy. Numbers were grouped, much like our current use of decimals and centesimals, except they worked in sixties rather than tens.

This setup was highly effective for their astronomical calculations, influencing timekeeping with 60 seconds in a minute and 60 minutes in an hour.
Symbolic Representation of Numbers
Both Egyptian hieroglyphics and Babylonian cuneiform relied on symbolic representation, replacing what we see as numerals today with icons or marks.

  • Hieroglyphics: Used pictorial symbols, each carrying a direct value.
  • Cuneiform: Used wedge-shaped marks in various combinations for numbers.


This symbolic approach had multiple purposes:
  • It facilitated understanding among those familiar with the symbols.
  • It added a layer of cultural significance, tying numerical representation to their artistry and communication methods.
These ancient systems were foundational to how we perceive numbers today, marrying visual artistry with mathematics.

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Most popular questions from this chapter

In the Babylonian system, multiply 25 by 1,04 and 18 by 1,21 . Divide 50 by 18 and 1,21 by 32 (using reciprocals). Use our standard multiplication algorithm modified for base \(60 .\)

Use the assumed Babylonian square root algorithm of the text to show that \(\sqrt{3} \approx 1 ; 45\) by beginning with the value 2 . Find a three- sexagesimal-place approximation to the reciprocal of \(1 ; 45\) and use it to calculate a three-sexagesimalplace approximation to \(\sqrt{3}\).

Show that the solution to the problem of dividing 7 loaves among 10 men is that each man gets \(\overline{\overline{3}} \overline{30}\). (This is problem 4 of the Rhind Mathematical Papyrus.)

Problem 72 of the Rhind Mathematical Papyrus reads " 100 loaves of pesu 10 are exchanged for loaves of pesu 45 . How many of these loaves are there? The solution is given as, "Find the excess of 45 over \(10 .\) It is 35 . Divide this 35 by 10. You get \(3 \overline{2}\). Multiply \(3 \overline{2}\) by 100. Result: 350. Add 100 to this 350 . You get 450 . Say then that the exchange is 100 loaves of pesu 10 for 450 loaves of pesu \(45 . "^{18}\) Translate this solution into modern terminology. How does this solution demonstrate proportionality?

Calculate a quantity such that if it is taken two times along with the quantity itself, the sum comes to 9. (problem 25 of the Moscow Mathematical Papyrus)
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