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Chapter 4: Problem 81

Why is a symbol for zero needed in a positional system?

Short Answer

Expert verified
In a positional system, a symbol for zero is needed as a placeholder to indicate an empty place, thereby maintaining the accurate positions and hence, values, of other digits in a number.

Step by step solution

01

Explain positional system

A positional system is a method of representation or encoding numbers. It uses a limited set of symbols, each of which has a specific place value. The position of the specific symbol is used to denote its value. For example, in the decimal number system, which is a positional system, the number 543 represents 5 hundreds (5 x 100), 4 tens (4 x 10), and 3 units (3 x 1). This due to the values of the positions or places.
02

Introduce the concept of zero

Zero is a numeral which plays a crucial role in positional systems. It allows us to differentiate between numbers that would otherwise look the same. For instance, in a system without zero, a number like 504 would look like 54 because there would be no way to represent the absence of tens, which could cause confusion or inaccuracies.
03

Describe the significance of zero

In a positional system, zero serves as a placeholder that indicates an empty place, preserving the positions of other digits in a number. For instance, in the number '205', the zero denotes that there are no tens. If this zero was not present, the number would be read as '25', which is incorrect. Thus, the symbol for zero is necessary to represent numbers accurately in a positional number system.

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Most popular questions from this chapter

In Exercises 1-12, add in the indicated base. $$ \begin{array}{r} 645_{\text {seven }} \\ +324_{\text {seven }} \\ \hline \end{array} $$

In Exercises 19-32, mentally convert each base ten numeral to a numeral in the given base. 25 to base \(\operatorname{six}\)

In Exercises 33-48, convert each base ten numeral to a numeral in the given base. 23 to base two

In Exercises 19-32, mentally convert each base ten numeral to a numeral in the given base. 3 to base two

In Exercises 13鈥�24, subtract in the indicated base. $$ \begin{array}{r} 32_{\text {four }} \\ -13_{\text {four }} \\ \hline \end{array} $$
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