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T101

SECTION 3-1

NUMERATION SYSTEMS

14five |||||||||

Do you know what all the following have in common? ···· 1001two IX

They are all the number ______, but written in various numeration systems or bases. By comparing our numeration system (Hindu-Arabic or Base-10) with other ancient numeration systems and other bases we hope to gain a clearer picture of how and why we do whole-number computations the way we do.

I.

HINDU-ARABIC A.

(ALSO KNOWN AS BASE-10 OR DECIMAL SYSTEM)

(Let's Review)

HOW MANY NUMERALS ARE THERE? ________ LIST THE NUMERALS: ______________________________________ (This means that each place in any number represents a power of 10 thus each digit in the number represents a certain value, namely, the place value column times the digit's face value) Tens (101 = 10) 7 Units (100 = 1) 2

B.

THE BASE IS ________

Thousands (103 = 1000) 5

Hundreds (102 = 100) 3

For example, in the numeral 5,372, the 5 sits in the place value column of "______________________" thus the digit 5 represents the value of ______ times _______________ = _______________.

Similarly, The digit 3 represents the value of ______ times _______________ = _______________.

The digit 7 represents the value of ______ times _______________ = _______________.

The digit 2 represents the value of ______ times _______________ = _______________.

Using the powers of 10, write 5,372 in expanded form:

Using the powers of 10, write 12,098 in expanded form:

C.

BASE-10 BLOCKS (See your Activity Book for blocks to be cut out) Base-10 Blocks are used to represent our Base-10 numbers. 1. Use Base-10 blocks to represent the following numbers. Draw your representations below.

234

102

2.

Take 18 units, 11 longs, and 2 flats now trade to have the fewest possible pieces. How many? What would be the largest number of pieces?

II.

EGYPTIAN NUMERATION SYSTEM This numeration system is based upon tally marks. That is, | 1 || 2 ||| 3 |||| 4 ||||| 5 |||||| 6 ||||||| 7 |||||||| 8 ||||||||| 9

But after 9, they needed a new symbol that began "grouping" to represent certain sets of numbers. The next value was (heel bone) that represented 10. Egyptian Numeral | Description Vertical Staff Heel Bone Scroll Lotus Flower Pointing Finger Polliwog Astonished Man Hindu-Arabic Equivalent 1 10 100 1,000 10,000 100,000 1,000,000 Write using Egyptian Numerals: 304,536 The Egyptian system simply followed an additive property; that is, the value of the number was the sum of the face values of the numerals. Convert to Base 10:

III.

ROMAN NUMERATION SYSTEM Hindu-Arabic 1 5 10 50 100 500 1000 Roman Numerals can be combined by using: 1. Additive property VI = XVII = CCCXXVIII =

Roman Numeral

2.

Subtractive property If a lesser value symbol is to the left of a higher value symbol, the lesser is subtracted from the greater. In general, only one lesser symbol can be to the left of a greater symbol. This prevents having to write a symbol more than 3 times in a row. IV = IX = XL = XC =

CD =

CM =

CXLIV =

MCDLXXI =

3.

Multiplicative property A bar(s) is place of over the symbol(s) indicating you should multiply by 1000 thus being able to create very large numbers

V=

DCLIX =

V

=

CVII =

IV.

BASE 5 (ALSO KNOWN AS THE QUINARY SYSTEM) A. HOW MANY NUMERALS ARE THERE? ________ LIST THE NUMERALS: B. THE BASE IS ________ This means that the columns are powers of 5 (not 10), thus each digit in the number represents a certain value, namely the place value column times the digit's face value. Column Names: ______________________________________

In the numeral 132five , the 1 sits in the place value column of "_______________________" thus the digit 1 represents the value of ______ times _______________ = _______________. Similarly, The digit 3 represents the value of ______ times _______________ = _______________. The digit 2 represents the value of ______ times _______________ = _______________. Thus the value of 132five in Base 10 or Hindu Arabic is ________. Using powers of 5, write 132five in expanded form: Write the first 10 counting numbers in Base 5:

Write as a Base 10 number: 124five 1030 five Write as Base 5 number: 43

318

824

V.

BASE 2 (ALSO KNOWN AS THE BINARY SYSTEM) A. HOW MANY NUMERALS ARE THERE? ________ LIST THE NUMERALS: B. THE BASE IS ________ This means that the columns are powers of ________ (not 10), thus each digit in the number represents a certain value, namely the place value column times the digit's face value. Column Names: ______________________________________

In the numeral 1011two , the first 1 sits in the place value column of "______________________" thus the digit 1 represents the value of ______ times _______________ = _______________. Similarly, The digit 0 represents the value of ______ times _______________ = _______________. The second 1 represents the value of ______ times _______________ = _______________. The third 1 represents the value of ______ times _______________ = _______________. Thus the value of 1011two in Base 10 or Hindu Arabic is ________. Using powers of 2, write 1011two in expanded form:

Write the first 10 counting numbers in Base 2:

Write as a Base 10 number: 1101 two 10111 two Write as Base 2 number: 27

57

69

VI.

BASE 12 A.

(ALSO KNOWN AS THE DUODECIMAL SYSTEM)

HOW MANY NUMERALS ARE THERE? ________ LIST THE NUMERALS: ______________________________________________________

B.

THE BASE IS ________ This means that the columns are powers of ________ (not 10), thus each digit in the number represents a certain value, namely the place value column times the digit's face value. Column Names:

In the numeral 96T twelve , the 9 sits in the place value column of "_______________________" thus the digit 9 represents the value of ______ times _______________ = _______________. Similarly, The digit 6 represents the value of ______ times _______________ = _______________. The digit T represents the value of ______ times _______________ = _______________. Thus the value of 96Ttwelve in Base 10 or Hindu Arabic is ________. Using powers of 12, write 96Ttwelve in expanded form:

Write the first 25 counting numbers in Base 12:

Write as a Base 10 number: 1Ttwelve 36ETtwelve

Write as Base 12 number: 39

804

1277

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