Reference no: EM131016455

1. Write the number 999,999 in

(a) Egyptian hieroglyphics,

(b) Babylonian cuneiform,

(c) Ionian Greek numerals,

(d) Roman numerals,

In addition to writing on this sheet 999,999 in Babylonian cuneiform, record in your AnswerEntryForm notebook the sexagesimal expansion of 999,999. For an example, the sexagesimal expansion of 987, 654, 321 would be recorded as

myAnswer1b = {21, 25, 28,12,16,1}

(e) Chinese rod numerals,

(f)  Traditional Chinese numerals, and

(g) Mayan numerals.

In addition to writing on this sheet 999,999 in Mayan numerals, record in your An- swerEntryForm notebook both the Mayan-modified vigesimal expansion of 999,999 and the unmodified of the same number . For an example, the Mayan-modified vigesimal and the unmodified vigesimal expansion of 987, 654, 321 would be recorded as, respectively,

myAnswer1gMayan = {1, 4, 4,14,18, 2,17}

and

myAnswer1gVigesimal = {1,16,15,16,12,8,15}

2. With respect to each of the number systems below, identify its base (and any subbases), whether it is positional, whether it is ciphered, and whether it has a (true) zero or not

(a) Egyptian hieroglyphics,

(b) Babylonian cuneiform,

(c) Ionian Greek numerals,

(d) Roman numerals,

(e) Chinese rod numerals,

(f) Traditional Chinese numerals, and

(g) Mayan numerals.

3. Write the number 123,456,789 in

(a) sexagesimal (base-60) notation,

(b) hexadecimal (base-16) notation

(c) vigesimal (base-20) notation

As previously, provide in each part a legend of the symbols being used. Whereas in parts (a) and (c) use ordinary integers such as 24 or 17, clearly separated by spaces, in part (b) use the ordinary hexadecimal symbols. Explain clearly, completely, how you are deriving the different expressions. Don't just write the answer down, derive it!

Record in your AnswerEntryForm notebook the three expansions of 999,999 in the bases 60, 16, and 20. For an example, the hexadecimal expansion of 987, 654, 321 would be recorded as

myAnswer3b = {1,11,8, 6,14,13,10,3}

4. Find, in the Egyptian fashion, the quotient 184 ÷ 8. Explain fully what you are doing

5.    

(a) Represent 3/29 as a sum of distinct unit fractions using the splitting identity;

(b) Represent 7/29 as a sum of distinct unit fractions the Fibonacci unit-fraction algorithm;

(c) Represent 7/29 in sexagesimal (base-60) notation;

(d) Represent 7/29 in hexadecimal (base-16) notation;

(e) Represent 7/29 in vigesimal (base-20) notation.

Explain fully your solutions-an answer alone without sufficient commentary is of little point-value. In parts (a) and (b) write the sum in terms of decreasing unit fractions (those with smaller denominators precede larger denominators). Thus, for an example, write the unit-fraction decomposition of  2/5  as the sum 1/4 + 1/10 + 1/20. In respect of parts (c) through (e) recall that the fraction 7/29, like any rational number, repeats not only in base 10 but also in any base b, where b is any positive integer. For an example, here is the base-25 expansion of the same fraction 7/29:

{6, 0, 21,13,19, 20,17,{R,1, 7}}

What this means is that

Represent as many places in the base-b expansion of 7/29 until it begins to repeat and then express your answer as I have just done in respect of the base b = 25.

Finally, record in your AnswerEntryForm notebook the five representations. In parts (a) and (b) your answer would assume the form

myAnswer5a = {1/4, 1/10, 1/20}

if the fraction were 2/5, and in parts (c) - (e) it would assume the form

myAnswer5c = {6, 0, 21, 13, 19, 20, 17,{R, 1, 7}}

if the fraction 7/29 were to be represented in base-25 notation.