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Chapter 4: Numeration Systems

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

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Historical Numeration Systems

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

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Numeration Systems

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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Example: Counting by Tallying

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

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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Ancient Egyptian Numeration – Simple Grouping

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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Ancient Egyptian Numeration

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Example: Egyptian Numeral

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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Ancient Roman Numeration

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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Ancient Roman Numeration

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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Ancient Roman Numeration

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Example: Roman Numeral

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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Traditional Chinese Numeration – Multiplicative Grouping

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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Chinese Numeration

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Example: Chinese Numeral

Solution

7000

200

400

0 (tens)

1

80

Answer: 201

2

Answer: 7482

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More Historical Numeration Systems

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

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Positional Numeration

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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Positional Numeration

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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Positional Numeration

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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Hindu-Arabic Numeration – Positional

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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Hindu-Arabic Numeration

Hundred thousands

Millions

Ten thousands

Thousands

Decimal point

Hundreds

Tens

Units

7, 5 4 1, 7 2 5 .

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Babylonian Numeration

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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Example: Babylonian Numeral

Interpret each Babylonian numeral.

a) ‹‹‹▼▼▼▼

b) ▼▼‹‹‹▼▼▼▼ ▼

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Example: Babylonian Numeral

Solution

‹‹‹▼▼▼▼

Answer: 34

▼▼‹‹‹▼▼▼▼ ▼

Answer: 155

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Mayan Numeration

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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Example: Mayan Numeral

Write the number below in our system.

Solution

Answer: 3619

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Greek Numeration

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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Arithmetic in the Hindu-Arabic System

- Expanded Form
- Historical Calculation Devices

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Expanded Form

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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Example: Expanded Form

Write the number 23,671 in expanded form.

Solution

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Example: Expanded Form

Use expanded notation to add 34 and 45.

Solution

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Decimal System

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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Historical Calculation Devices

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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Abacus

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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Example: Abacus

Which number is shown below?

Solution

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

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Lattice Method

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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Example: Lattice Method

Find the product by the lattice method.

Solution

Set up the grid to the right.

7 9 4

3

8

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Example: Lattice Method

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

1

2

3

0

1 7 2

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Nines Complement

- 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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Example: Nines Complement Method

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

- General Base Conversions
- Computer Mathematics

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General Base Conversions

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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Powers of Alternative Bases

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Example: Converting Bases

Convert 2134five to decimal form.

Solution

2134five

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Calculator Shortcut for Base Conversion

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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Example: Calendar Shortcut

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

Solution

432134five

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Example: Converting Bases

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

1072 4

153 1

21 6

3 0

0 3

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Converting Between Two Bases Other Than Ten

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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Computer Mathematics

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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Example: Convert Binary to Decimal

Convert 111001two to decimal form.

Solution

111001two

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Example: Convert Hexadecimal to Binary

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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