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Chapter 4

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Chapter 4

Numeration Systems

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- 4.1 Historical Numeration Systems
- 4.2 More Historical Numeration Systems
- 4.3Arithmetic in the Hindu-Arabic System
- 4.4 Conversion Between Number Bases

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- Historical Numeration Systems

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- Basics of Numeration
- Ancient Egyptian Numeration
- Ancient Roman Numeration
- Classical Chinese Numeration

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The various ways of symbolizing and working with the counting numbers are called numeration systems. The symbols of a numeration system are called numerals.

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Tally sticks and tally marks have been used for a long time. Each mark represents one item. For example, eight items are tallied by writing the following:

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Counting by grouping allows for less repetition of symbols and makes numerals easier to interpret. The size of the group is called the base (usually ten) of the number system.

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The ancient Egyptian system is an example of a simplegrouping system. It uses ten as its base and the various symbols are shown on the next slide.

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2012 Pearson Education, Inc.

Write the number below in our system.

Solution

2 (100,000) = 200,000

3 (1,000) = 3,000

1 (100) = 100

4 (10) = 40

5 (1) = 5

Answer: 203,145

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The ancient Roman method of counting is a modified grouping system. It uses ten as its base, but also has symbols for 5, 50, and 500.

The Roman system also has a subtractive feature which allows a number to be written using subtraction.

A smaller-valued symbol placed immediately to the left of the larger value indicated subtraction.

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The ancient Roman numeration system also has a multiplicative feature to allow for bigger numbers to be written.

A bar over a number means multiply the number by 1000.

A double bar over the number means multiply by 10002 or 1,000,000.

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2012 Pearson Education, Inc.

Write the number below in our system.

MCMXLVII

Solution

M= 1000

CM= -100 + 1000

XL = -10 + 50

V= 5

I= 1

I= 1

Answer:

1000 + 900 + 40 + 5 + 1 + 1= 1947

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A multiplicative grouping system involves pairs of symbols, each pair containing a multiplier and then a power of the base. The symbols for a Chinese version are shown on the next slide.

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Interpret each Chinese numeral.

a)b)

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Solution

7000

200

400

0 (tens)

1

80

Answer: 201

2

Answer: 7482

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- More Historical Numeration Systems

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- Basics of Positional Numeration
- Hindu-Arabic Numeration
- Babylonian Numeration
- Mayan Numeration
- Greek Numeration

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A positional system is one where the various powers of the base require no separate symbols. The power associated with each multiplier can be understood by the position that the multiplier occupies in the numeral.

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In a positional numeral, each symbol (called a

digit) conveys two things:

1.Face value – the inherent value of the symbol.

2.Place value – the power of the base which is associated with the position that the digit occupies in the numeral.

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To work successfully, a positional system must have a symbol for zero to serve as a placeholder in case one or more powers of the base is not needed.

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One such system that uses positional form is our system, the Hindu-Arabic system.

The place values in a Hindu-Arabic numeral, from right to left, are 1, 10, 100, 1000, and so on. The three 4s in the number 45,414 all have the same face value but different place values.

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Hundred thousands

Millions

Ten thousands

Thousands

Decimal point

Hundreds

Tens

Units

7, 5 4 1, 7 2 5 .

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The ancient Babylonians used a modified base 60 numeration system.

The digits in a base 60 system represent the number of 1s, the number of 60s, the number of 3600s, and so on.

The Babylonians used only two symbols to create all the numbers between 1 and 59.

▼ = 1 and ‹ =10

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Interpret each Babylonian numeral.

a) ‹‹‹▼▼▼▼

b) ▼▼‹‹‹▼▼▼▼ ▼

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Solution

‹‹‹▼▼▼▼

Answer: 34

▼▼‹‹‹▼▼▼▼ ▼

Answer: 155

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The ancient Mayans used a base 20 numeration system, but with a twist.

Normally the place values in a base 20 system would be 1s, 20s, 400s, 8000s, etc. Instead, the Mayans used 360s as their third place value.

Mayan numerals are written from top to bottom.

Table 1

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Write the number below in our system.

Solution

Answer: 3619

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The classical Greeks used a ciphered counting system.

They had 27 individual symbols for numbers, based on the 24 letters of the Greek alphabet, with 3 Phoenician letters added.

The Greek number symbols are shown on the next slide.

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Table 2

Table 2

(cont.)

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Interpret each Greek numeral.

a)ma

b)cpq

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Solution

a) ma

b) cpq

Answer: 41

Answer: 689

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- Arithmetic in the Hindu-Arabic System

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- Expanded Form
- Historical Calculation Devices

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By using exponents, numbers can be written in expanded form in which the value of the digit in each position is made clear.

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Write the number 23,671 in expanded form.

Solution

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For all real numbers a, b, and c,

For example,

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Use expanded notation to add 34 and 45.

Solution

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Because our numeration system is based on powers of ten, it is called the decimal system, from the Latin word decem, meaning ten.

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One of the oldest devices used in calculations is the abacus. It has a series of rods with sliding beads and a dividing bar. The abacus is pictured on the next slide.

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Reading from right to left, the rods have values of 1, 10, 100, 1000, and so on. The bead above the bar has five times the value of those below. Beads moved towards the bar are in “active” position.

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Which number is shown below?

Solution

1000 + (500 + 200) + 0 + (5 + 1) = 1706

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The Lattice Method was an early form of a paper-and-pencil method of calculation. This method arranged products of single digits into a diagonalized lattice.

The method is shown in the next example.

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Find the product by the lattice method.

Solution

Set up the grid to the right.

7 9 4

3

8

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Fill in products

7 9 4

3

8

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Add diagonally right to left and carry as necessary to the next diagonal.

1

2

3

0

1 7 2

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1

2

3

0

1 7 2

Answer: 30,172

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- Used for subtracting
- We first agree that the nines complement of a digit n is 9 – n. For example, the nines complement of 0 is 9, of 1 is 8, of 2 is 7, etc.
- Step 1 – Align the digits as in a standard subtraction problem.
- Step 2 – Add leading zeros, if necessary, in the subtrahend so that both numbers have the same number of digits.
- Step 3 – Replace each digit in the subtrahend with its nines complement, and then add.
- Step 4 – Finally, delete the leading digit (1), and add 1 to the remaining part of the sum.

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Use the nines complement method to subtract 2803 – 647.

Solution

Step 1 Step 2 Step 3 Step 4

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- Conversion Between Number Bases

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- General Base Conversions
- Computer Mathematics

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We consider bases other than ten. Bases other than ten will have a spelled-out subscript as in the numeral 54eight. When a number appears without a subscript assume it is base ten. Note that 54eight is read “five four base eight.” Do not read it as “fifty-four.”

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2012 Pearson Education, Inc.

Convert 2134five to decimal form.

Solution

2134five

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To convert from another base to decimal form: Start with the first digit on the left and multiply by the base. Then add the next digit, multiply again by the base, and so on. The last step is to add the last digit on the right. Do not multiply it by the base.

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Use the calculator shortcut to convert 432134five to decimal form. (Can put the formula below directly into Excel!)

Solution

432134five

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Convert 7508 to base seven.

Remainder

Solution

Divide by 7, then divide the resulting quotient by 7, until a quotient of 0 results.

From the remainders (bottom to top) we get the answer:

7508 = 30614seven

7508

10724

1531

216

30

03

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Many people feel the most comfortable handling conversions between arbitrary bases (where neither is ten) by going from the given base to base ten and then to the desired base.

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There are three alternative base systems that are most useful in computer applications. These are binary (base two), octal (base eight), and hexadecimal (base sixteen) systems.

Computers and handheld calculators use the binary system.

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Convert 111001two to decimal form.

Solution

111001two

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Convert 8B4Fsixteen to binary form.

Solution

Each hexadecimal digit yields a 4-digit binary equivalent.

8 B 4 Fsixteen

1000 1011 0100 1111two

Combine to get

8B4Fsixteen = 1000101101001111two.

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