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 Bases
Our base 10:
The column headings
are:
1000
100
10
1
Mayan base 20:
The row headings are:
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
What numbers are shown?
8000
8000
8000
8000
400
400
400
400
20
20
20
20
1
1
1
1
Remember:
is 0
is 1
and
is 5
Write this
Write the following using Mayan numbers.
Use the row headings to help you if you need to.
• 21
• 63
• 40
• 97
• 100
• 372
Writing Mayan Numbers: answers
• 21 would be:
• 63 would be:
• 40 would be:
• 97 would be:
• 100 would be:
• 372 would be:
Larger Mayan Numbers
To 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
(4 left)
4 x
20 = 320
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
18 uinals
20 tuns
20 katuns
=
=
=
=
1 uinal
1 tun
1 katun
1 baktun
=
=
=
=
20 days
360 days
7,200 days
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 =
160 =
17 =
1377_
2000 =
000 =
12 =
• 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 Numbers
Workbook 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.