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Chapter 4: Problem 70

Describe how to change a numeral in a base other than ten to a base ten numeral.

Short Answer

Expert verified
The numeral \(1011_2\) converts to the numeral \(11_{10}\) in base ten.

Step by step solution

01

Comprehend the numeral system.

It is crucial to understand that a numeral system is a way of expressing numbers using a consistent set of symbols. A base, or radix, is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, base 10 (decimal) uses the digits from 0 to 9, base 2 (binary) uses 0 and 1, and base 16 (hexadecimal) uses 0 to 9 and A to F.
02

Identify the Numerals

In order to solve the exercise and change a numeral in a base other than ten to a base ten numeral, you need to know the numeral you are starting with and the base it's in. For example, let's select the numeral \(1011_2\), which is in base 2.
03

Convert to Base Ten

To convert a numeral from any base to base ten, sum the products of each digit by the base raised to the power of the digit's index, starting the count from zero at the rightmost digit. The conversion of \(1011_2\) to base 10 happens as follows: \[(1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) = 8 + 0 + 2 + 1 = 11_{10}\] Hence, \(1011_2 = 11_{10}\).

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Most popular questions from this chapter

In Exercises 1-12, add in the indicated base. $$ \begin{array}{r} 6784_{\text {nine }} \\ +7865_{\text {nine }} \\ \hline \end{array} $$

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