Assignment #2 – Counting Systems and Calendars



Assignment #2 – Counting Systems and Calendars

Counting Systems

Modern science uses a base ten counting system based on Arabic characters, with a zero number. (Probably derived from the fact that we have ten fingers.) However through time there have been a variety of different counting systems with different bases. Our hour system for example does not follow the base ten but rather a base 60 system inherited from the Babylonian system of counting. (This is also why there are 360 degrees in a circle.) Here we look at counting in other systems such as the Mayans, and Babylonians used.

The Mayan number system is based not on tens, but on twenties, and does have a zero character (this is not always the case). Looking at the diagram on page 108 of ‘Stairways to the Stars’ we have the counting systems laid out, with symbols for each of the numbers. Similar to our current counting system, position indicates how many of each unit you need.

The Babylonians were among the first to have a placeholder counting system as we use today, and our modern numbering system does. For example in the number 120 – we know we have one hundred, two tens and no ones. Not all cultures developed this type of numbering system. The Egyptians for example used symbols for numbers, and just wrote them as many times as was needed. This is closer to the roman numeral system – where I is one and three is III, (three ones), X is ten and XXX is thirty etc.

Calendars and Time Keeping

Currently modern western society uses the Gregorian calendar – 365 days a year, with leap years every four years. This is based on the movement of the sun and the dates of the equinoxes and solstices. However, there are other movements that were used to base a calendar on, the most common being a lunar calendar, based on either the phases of the moon (synodic period – about 30 days), or the sidereal period (about 27 days). You can also use the rising of a star as the beginning of your year. Many ancient cultures had complex calendars that used combinations of these cycles. Even our modern calendar does as months on based on a lunar cycle of 30 days, while the week is basically the length of each phase (new, first quarter, full, last or third quarter).

The length and days of the week are actually derived from the names and number of the visible bodies in our solar system - Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn ( 7 naked eye visible objects). The English names currently in use derive for the most part from the Norse mythology – Woden becomes Wednesday, Friga comes from Friday etc. (see also weekday names in the images pages on the Website)

Because the vernal equinox year is 365.2425 years long, one leap year every four years with exceptions at 3 out of four centuries is still not as accurate as it could be. More accurate would be to insert 8 leap days every 33 years. (this calendar is in use in some middle eastern nations)

Perpetual Calendars First Weekday

|non-Leap Years |Leap Years |Starting Day |non-Leap Years |Leap Years |Starting Day |

|Calendar 0 |Calendar 7 |Sunday |Calendar 4 |Calendar 11 |Thursday |

|Calendar 1 |Calendar 8 |Monday |Calendar 5 |Calendar 12 |Friday |

|Calendar 2 |Calendar 9 |Tuesday |Calendar 6 |Calendar 13 |Saturday |

|Calendar 3 |Calendar 10 |Wednesday  | | | |

|1901 - 2 |1911 - 0 |1921 - 6 |1931 - 4 |1941 - 3 |

| | | | |[pic] |

| | |[pic] |[pic] |[pic] |

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Calendars

1. When looking at a perpetual calendar, why are there fourteen possibilities for different types of years?

2. Give the starting weekday for the following years are they leap years or not?

a. 2005

b. 2068

c. 2100

3. If humans are going to colonize Mars we will need to develop a Mars calendar. Mars has a year of 686.98 Earth days of 24 hours. However Mars has a day length of 24.6229 hours.

a. How many Martian days are there in a year?

b. How often would Mars need a leap year? (How many times the decimal of a year will give you a whole number? You may find that you need to add for example 5 days every 7 years to make things work out)

c. If Mars had a moon that orbited with a period of 42.40 Martian days, how many Martian full moons would there be in a year? How many extra days would there be in a solar year as compared to a lunar year?

d. How would you rearrange the number of days in the Martian months to match the number of days in a Martian Year?

Name: Student Number: .

Answer Sheet - Exercise # 3

Counting

1. Mayan Number – each column represents one number

|A |B |C |D |E |

| | | | | |

| | | | | |

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| | | | | |

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2. Normal Numbers

a.

b.

c.

d.

e.

Calendars

3. Explanation for number of years?

4. Starting day of week, is it a leap year?

| | | |

| | | |

| | | |

5. Martian Calendar

a. Martian Days in a year

b. Cycle of Martian Leap Years

c. Martian months in a year? Extra days?

d. Suggested adjustment of month lengths.

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