Chapter 4.CR, Problem 1CR

To determine

To explain:

The meaning when we say that a numeration system is positional with an examples.

Answer to Problem 1CR

Solution:

When the numeration system is positional, then the symbols of numerals should be placed in its position for not to change the original value of the number.

Explanation of Solution

Consider the following examples of in positional system.

The above given numeral is the system of Babylonian numeration.

The following table lists the value of each selected numeral of .

Selected Numerals (Egyptian numeration)	Equivalent numeral in decimal system
	1
	10

The position of the numeral is right of the numeral .

Thus, the given numeral must be greater than 60.

Hence,

$=\left(1\times 60\right)+10+1$

$=60+10+1$

$=71$

The above given numeral is the system of Babylonian numeration.

The following table lists the value of each selected numeral .

Selected Numeral (Babylonian numeration)	Equivalent numeral in decimal system
	1
	10

The position of the numeral is left of the numeral and also to the right of the same .

Thus, the given numeral must be greater than 60.

Also here we have numeral to the left of the second 60-group.

Thus,

$=\left(10+1\right)\times 60+10+1$

$=11\times 60+10+1$

$=660+10+1$

$=671.$

XLVIII

The above given numeral is in the system of Roman numeration.

The following table lists the value of each selected numeral of $\overline{\text{DL}}.$

Selected Numeral (Roman Numeration)	Equivalent numeral in decimal system
X	10
L	50
V	5
I	1

Thus,

$=\text{XLVIII}$

$=\left(50-10\right)+5+1+1+1$

$=40+5+3$

$=48$

DCCIX

The above given numeral is in the system of Roman numeration.

The following table lists the value of each selected numeral of CD.

Selected Numeral (Roman Numeration)	Equivalent numeral in decimal system
D	500
C	100
I	1
X	10

Thus,

$=\text{DCCIX}$

$=500+100+100+\left(10-1\right)$

$=700+9$

$=709$

In the above examples changing the position of the symbol changes the value of the numeral.

Therefore, when the numeration system is positional, then the symbols of numerals should be placed in its position for not to change the original value of the number.