What have the Mayans done for us?

The Mayan civilization spread all over south-eastern Mexico, Guatemala, Belize, and Honduras between 2500 B.C. and A.D. 250. They are well known for their mathematical and astronomical systems.

Mayan Mathematics• Inscriptions show the Mayans

were working with calculations up to hundreds of millions.

• They produced extremely accurate astronomical data from naked eye observations.

• The Mayans measured the length of the year to a high degree of accuracy, but approximated it to 360 days.

Mayan Number System• The Mayans used a number system based on

1s, 5s and 20s. • They were one of the only ancient civilizations to

use a zero.• The Mayans devised a counting system that was

able to represent very large numbers by using only 3 symbols: a dot, a bar, and a glyph for zero, usually a shell.

Mayan Number symbols 0 -19• Zero is represented by a

shell; 1 to 4 are represented by dots.

• Multiples of five are represented by lines, with extra dots being added to complete the numbers as shown.

• Can you work out the missing diagrams?

Writing Bigger Numbers• Our own number system is base 10, which

means that we have 9 ‘symbols’ (1,2,3,4,5,6,7,8,9) plus a zero.

• The Mayan number system had 19 symbols (as on the previous slide) plus a shell for zero.

• When writing numbers, once we get to ‘9’ we then have to move across to the next column. We write a ‘one’ followed by a ‘zero’ to show that we have moved across – zero is a ‘place-holder’

Writing Bigger Numbers• The Mayans used a similar system using their

19 symbols and then moving to the next section and putting a zero (represented by the shell) as a placeholder.

• Another difference is that they used rows instead of columns, starting from the bottom and working upwards.

Writing Bigger Numbers• In base 10, the headings are 100, 101, 102, 103,

104 etc.• In base 20 the headings are 200, 201, 202, 203,

204 etc.• What are these values?

Number BasesOur base 10:

The column headings are:

Mayan base 20:

The row headings are:

1000 100 10 1

8000

400

20

1

It will help initially to see the row headings, but they would not normally be shown… just as our young children use 1, 10 and 100 as column headings when they begin writing numbers.

Writing Mayan Numbers

8000

400

20

1

8000

400

20

1

8000

400

20

1

8000

400

20

1

What numbers are shown?

Remember: is 0 is 1 and is 5

Write thisWrite the following using Mayan numbers.

Use the row headings to help you if you need to.

• 21

• 40

• 100

• 63

• 97

• 372

Writing Mayan Numbers: answers

• 21 would be:

• 40 would be:

• 100 would be:

• 63 would be:

• 97 would be:

• 372 would be:

Larger Mayan NumbersTo write larger numbers start with the highest row that can be subtracted from the number you are trying to write.

Example: Writing 5124.

‘8000’ is too big, so start with as many 400s as possible, then work down with what’s left for the 20s and 1s.

5124 = 12 x 400 = 4800

(324 left) 16 x 20 = 320

(4 left) 4 x 1 = 4

Problem 1• How would you write 1377 in Mayan numbers?• Now try writing 2012.

Other Mayan Number Systems• The Mayan had a second

Number System, used for dating buildings and on Calendars, etc.

• This would be a more formal system, rather than a number system used for calculation.

Mayan Calendar• Maya dates combined at least two calendars -

one, the ‘Calendar Round’, covering 365 days and the other 260 days, such that every day had two names, which reset every 52 years.

• The Maya also used a “Long Count" system of 187,2000 days that added a numeral at the end of a cycle to keep a constant count of years.

• It is important to note that the Long Count's version of a year, the tun, is only 360 days, not the solar count of 365.

Mayan Calendar• The basic unit for the Mayan calendar is the kin.

20 kins = 1 uinal = 20 days 18 uinals = 1 tun = 360 days 20 tuns = 1 katun = 7,200 days 20 katuns = 1 baktun = 144,000 days

• Every date expressed in long count terms contained five numerals, that is, the number of baktuns, katuns, tuns, uinals (or winals) and kins elapsed from the "beginning of time", according to the Maya system.

Problem 2• How many (Long Count) years

in a baktun?

• There are 13 baktuns in a “great cycle”.

• How many years is this?

Long Count Calendar• Starting at ‘year zero’ – the very beginning of a Long

Count period - the read-out of the calendar was set at: 0.0.0.0.0.

• When each value was numerically accomplished to its maximum, it would then reset to ‘0’ and the total would be carried forward into the next time cycle to its left.

• The beginning of the current cycle corresponds to August 13, 3114 B.C. on the Gregorian calendar.

• This cycle is due to end on 13.0.0.0.0, the end of the 13th baktun.

Problem 3• These are typically recorded by archaeologists

translating the ancient Maya script, like this:baktun.katun.tun.winal.kin

• Can you work out what date is represented by: 12.19.19.17.19?

Remember:

20 kins = 1 uinal = 20 days 18 uinals = 1 tun = 360 days 20 tuns = 1 katun = 7,200 days 20 katuns = 1 baktun = 144,000 days

Mayan Apocalypse?• The Mayan calendar finishes

one of its great cycles in December 2012, which has fuelled countless theories about the end of the world at 11:11 on December 21, 2012.

Not the End of the World• Just as the calendar you have on your wall does

not cease to exist after December 31, the Mayan calendar does not cease to exist on December 21, 2012.

• This date is the end of the Mayan long count period but then another long count period begins for the Mayan calendar.

Teachers Notes: structuring the work

The work on Mayan number systems and calendars could be split into two sections according to need and time, using sections 1 to 15 and then 16 to 23 at a later time. The information from the first section is not required in order to complete the second section.

Both sections reinforce working with number, and although working in other number bases is currently not part of most GCSE syllabuses, working on this type of activity often helps pupils to better understand and appreciate the structure of base 10.

The activities are best suited to a combination of short teacher-lead information sessions to help pupils to understand the systems, followed by paired working on the problems.

Teachers Notes: short answers from slides

Slide 9: values for base 10 are 1 10 100 1000 10000

values for base 20 are 1 20 400 8000 160000

Slide 11: numbers shown are 20 410 900 551

Answers

• Problem 1 1200 = 2000 = 160 = 000 = 17 = 12 =1377_

• Problem 2 400 years; 5200 years

• Problem 3 12 baktun, 19 katun, 19 tun, 17 winal, and 19 kin, or

December 20, 2012.

External Resources

• Cracking the Maya Code

Students see how scientists began to unravel the meaning of Maya glyphs and then determine their own birth date using the Maya Long Count calendar system.

• How to Calculate with Mayan NumbersWorkbook for students to practice addition, subtraction, multiplication, division, and square roots using Mayan numbers.

• The Exploratorium’s Mayan Calendar In this 1-2 hour activity, students will learn about the two calendars the Maya used, and solve the problem of how often the two cycles coincided, by making and rotating gears, and by using prime numbers and smallest common multiples.