Math 365 Lecture Notes - © S. Nite 1/14/2012 Page 1 of 4

Section 2-1

2.1 Numeration Systems

Hindu-Arabic Numeration System

The system we use today was developed by the Hindus and taken to Europe by the

Arabs. Two important properties:

• All numerals are construction from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

• Place value is based on powers of the number base, 10.

Example:

Write 6,493 in expanded form.

Represent 2,136 with base ten blocks.

What is the fewest number of pieces you can receive in a fair exchange for 12 flats,

17 longs, and 14 units?

Tally Numeration System

In a tally system, there is a one-to-one correspondence between the marks and the

items being counted. Grouping is used to make it easier to read.

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Section 2-1

Egyptian Numeration System

The Egyptian numeration system improved on the tally system by developing a

grouping system for certain sets of numbers, making it easier to read. It had an

additive, rather than place-value property.

Example:

Represent 1254 in the Egyptian system.

Babylonian Numeration System

The Babylonian system was a place value system, with base 60.

▼ = 1 < = 10

Examples:

Find the value of the Babylonian number:

▼▼▼ <<

1 060 60

+

Find the value of the Babylonian number:

▼▼▼ ▼▼ <<▼

2 1 06 0 6 0 6 0

+ +

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Section 2-1

Mayan Numeration System

The Mayan system used a place-value system, with only three symbols. They

introduced the symbol for zero.

The system was a modified base 20 system, using 360 (approximate number of

days in a year) where one would expect 400. For an explanation of why they chose

this number system go to the following website http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Mayan_mathematics.html

Roman Numeration System

The Roman system uses additive, subtractive, and multiplicative properties.

Examples:

What does CLXIV represent?

Represent 734 in the Roman system.

Other Number Base Systems

The Luo peoples of Kenya used a base five system, which can be modeled by

counting with only one hand.

Examples:

Convert 11244five to base 10.

What number follows 44five?

Write 724 in base 5.

=1

=5

=0

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Section 2-1

The binary system (base two) is used in some aboriginal tribes. It is important in

computers because the presence or absence of an electrical signal requires two

digits.

Examples:

Write the first ten natural numbers in base 2.

Write 37 in base two.

Convert 10101two to base ten.

Base twelve (duodecimal) is based on dozens, which is used for purchasing eggs,

pencils, etc.

Examples:

In base twelve, what five numbers come after 19?

Convert T2Etwelve to base ten.

Write 1275 in base twelve.