Lesson 2

Lesson 3.1F’08

Objectives: 1. To become familiar with Egyptian,

Mayan, and Babyl= onian number systems.

2. To learn to use the Ro= man Numeration

system.

3. To convert from base ten to other bases

and from o= ther bases to base 10.

Principals and Standar= ds page 153

Numerals- are symbols like 2 <= span style=3D'mso-spacerun:yes'> 4 6 etc.

In our Hindu Arabic system writing numbers greater than 9 requires a numeration system.

Numeration system- is a collection of properties and symbols agreed upon to represent numbers systematically.

In the Hindu Arabic system: (developed by= the Hindus and transported to Europe by the Arabs).

1.      All numerals are const= ructed from ten digits.

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

&n= bsp; base of the system.

Each place represents a power of ten.

Place Value

Millions=

Hundred thousands

Ten-thousands<= /o:p>

thousands

hundreds=

tens

Units

Face Value

5

7

8

3

4

2

1

Place value= -names its position in the numeral.

Face value<= /b> -the digit itself- te= lls how many groups of ten

    (or ten to a power) ar= e indicated.

Example= : Write 3768 in expand= ed form.

3768 =3D 3 x 10³ + 7 x 10² + 6 x 10 + 8 x 10⁰

Definiti= on of aⁿ

If a is any number and n is any natural number, then

        aⁿ =3D a a a a a …a   n factors<= /p>

TRY THIS

Write 98,759 in expanded form.

Base-ten blocks—units, longs, flats and blocks

1,      10,   100,       1000=

These can be used to teach place value.

Example: What is the fewest number of piece= s you can receive in a fair exchange for:

1. 15 units   11 longs    2 flats

TRY THESE: : What is the fewest number of piece= s you can receive in a fair exchange for:

1. 22 units   15 longs    4 flats

2. 7 units   20 longs    3 flats

Other Numeration Systems

Tally Numeration Syste= m-uses single strokes or tally marks to represent each object being counted.

l, ll, lll, llll, lllll, llllll-- Ther= e is a one-to-one correspondence between the marks and the items being counted.

Egyptian Numeration Sy= stem-used tally marks for numbers 1-9. They developed a grouping system to represent certain sets of numbers. Look at the table page 158 of your book.

Egyptian numeral hieroglyp= hs

1
10⁰

10
10¹

100
10²

1,000
10³

10,000
10⁴

100,000 10⁵

1,000,000=
10⁶

10,000,00= 0
10⁷

              &= nbsp;        represents

Babylonian Numeration = System- Used a place value system. Numbers greater than = 59 were represented by repeated groups of 60.

Babylonian Numeral

Hindu-Arabic

       &nbs= p;

       &nbs= p; 1

       &nbs= p;       <

       &nbs= p; 10

<= < &n= bsp;            = ;   &nb= sp;                   represents

<= <        =           =             &nb= sp;         represents

        =    < <                &= nbsp;            represents

Try These:

1. Use the Egyptian system to represe= nt 11,321.

2. Use the Babylonian system to repre= sent 12,345.

Mayan Numeration Syste= m-used a symbol for zero. This system used only t= hree symbols. This system is almost= a base-twenty system.   Ins= tead of place values of

1, 20, 20², 20³, 20⁴, … the Mayans used=

1, 20, 1820, 1820², <= span style=3D'font-size:18.0pt;mso-bidi-font-size:12.0pt'>1820³, 1820⁴, …<= o:p>

The symbols used by t= he Mayans were ( page 160)

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

Mayan Numerals

Hindu-Arabic

  &= nbsp;

       &nbs= p; 0

       &nbs= p;

       &nbs= p;   1

       &nbs= p;

       &nbs= p;   5

Roman Numeration System-(need to know)-We can see this system today. It can be seen on cornerstones, on opening pages of books, etc. = The Roman system uses only a few symbols. (page 161)

character

numerical value

      I

1

      V

5

      X

10

      L

50

      C

100

      D

500

      M

        1000

To avoid repeating a = symbol more than three times a subtractive property was introduced. IV represents 5 - 1 or 4. IX represents 10 -1 or 9. Look at the following table with g= ives all subtraction possible.

Roman Numeral<= /o:p>

Hindu-Arabic

       &nbs= p;   IV

       &nbs= p; 5 - 1 =3D 4

       &nbs= p; IX

       &nbs= p; 10 - 1 =3D 9

       &nbs= p; XL

       50= - 10 =3D 40

       &nbs= p; XC

      100 - 10= =3D 90

        &nbs= p; CD

    500 - 100 =3D 400

       &nbs= p; CM

1000 - 100 =3D 900

A bar is placed over a Roman numeral to indicate that it is multiplied by 1000. Two bar o= ver a Roman numeral indicates that it is multiplied by 1000² etc.=

Example:   represents (4) 1000 or 4000=

            =     represents (3) 1000² or 3,000,000

Try These: Convert each Roman Numeral to Hindu Arabic

1.      CCLXXI

2.      MDVIII

3.      CCXLIX

4.      CLIX

OTHER BASES=

1. Base five- digits 0, 1, 2, 3, 4

Count in base 5- use the base five pieces

2. Base two-digits 0,1

C= ount in base two

3. Count in base 12 ( t =3D 10, e =3D= 11)

Example: Express each number in base 10

1. 123_five

2. 1234_eight

TRY THESE: Express each number in= base 10

1.&n= bsp;     231_fiv=
e

2. 426_seven

Example: Convert from base ten to the indic= ated base.

1. 53 to base two

2. 395 to base eight_<=
/sub>

TRY THESE: Convert from base ten to the indicated base=

1.      426 to base seven=

2. 43 to base two

Class work Worksheet 3.1

Homework Math XL section 3.1 and bookwork #1= ab, 2a-c, 8, 9, 10, 11, 12, 16, 17, 18a, 20, 23, 25, 30, 31, 32, 33<= /span>

Due By ________________________

Place Value	Millions=	Hundred thousands	Ten-thousands<= /o:p>	thousands	hundreds=	tens	Units
Face Value	5	7	8	3	4	2	1

Egyptian numeral hieroglyp= hs

1 10⁰	10 10¹	100 10²	1,000 10³	10,000 10⁴	100,000 10⁵	1,000,000= 10⁶	10,000,00= 0 10⁷

Babylonian Numeral	Hindu-Arabic
&nbs= p;	&nbs= p; 1
&nbs= p; <	&nbs= p; 10

Mayan Numerals	Hindu-Arabic
&= nbsp;	&nbs= p; 0
&nbs= p;	&nbs= p; 1
&nbs= p;	&nbs= p; 5

Roman Numeral<= /o:p>	Hindu-Arabic
&nbs= p; IV	&nbs= p; 5 - 1 =3D 4
&nbs= p; IX	&nbs= p; 10 - 1 =3D 9
&nbs= p; XL	50= - 10 =3D 40
&nbs= p; XC	100 - 10= =3D 90
&nbs= p; CD	500 - 100 =3D 400
&nbs= p; CM	1000 - 100 =3D 900