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Chapter 5: Problem 11

Express each binary number in decimal. $$ 1001 $$

Short Answer

Expert verified
The decimal equivalent of the binary number 1001 is 9.

Step by step solution

01

Identify the Position of Each Digit

Reading from right to left, the binary digits hold the following positions: 0, 1, 2, and 3.
02

Calculate the Value for Each Digit

Calculate the value for each digit of the binary number, which is the digit multiplied with 2 raised to the power of its position.\n\n Thus, the values would be: \( 1 \times 2^3 \), \( 0 \times 2^2 \), \( 0 \times 2^1 \), and \( 1 \times 2^0 \).\n Which simplifies to: 8 , 0, 0 and 1.
03

Sum the Values

The equivalent decimal number is the sum of these values, i.e., 8 + 0 + 0 + 1 = 9.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decimal System
The decimal system is the number system most of us use in our daily lives. It is based on ten unique digits, ranging from 0 to 9. This system works on the principle of powers of ten.
Each position in a decimal number represents a specific power of ten based on its place. For instance, in the number 345, the digit 5 is in the 'ones' place, the digit 4 is in the 'tens' place, and the digit 3 is in the 'hundreds' place.
  • The 'ones' place signifies \( 10^0 \)
  • The 'tens' place signifies \( 10^1 \)
  • The 'hundreds' place signifies \( 10^2 \)
Every shift to the left in a decimal number increases the place value tenfold. It's this simple and intuitive structure that makes the decimal system so universally adopted.
Binary Numbering System
Unlike the decimal system, the binary numbering system is based on just two digits: 0 and 1. This system is the backbone of how computers store and process data.
In binary, each position of a digit represents a power of two. For example, in the binary number 1001, we can break down each digit and its positional value as follows:
  • The far-right '1' is in the \( 2^0 \) position
  • The next '0' is in the \( 2^1 \) position
  • The third '0' is in the \( 2^2 \) position
  • The far-left '1' is in the \( 2^3 \) position
To find a decimal equivalent in binary, simply multiply each binary digit by 2 raised to the power of its position, starting from the right and moving left, then sum these values.
Number Position Value
The concept of number position value is central in both binary and decimal systems. It helps us understand how individual digits contribute to the total value of a number based on their position. In a binary number, each digit's position determines the power of two it represents, while in a decimal number, each digit鈥檚 position signifies a power of ten.
When converting a binary number to decimal, you:
  • Start from the rightmost digit
  • Identify its position (0, 1, 2, ...)
  • Multiply the digit by 2 raised to this position
  • Add up all these products for the total decimal value
This process highlights how the position of each digit gives it a unique value, making it crucial for numerical systems.

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Most popular questions from this chapter

Find the greatest common divisor of each pair of integers. $$ 331,993 $$

Show that \(\operatorname{gcd}(n, \phi)=1,\) and find the inverse s of \(n\) modulo \(\phi\) satisfying \(0

Use the following notation and terminology. We let \(E\) denote the set of positive, even integers. If \(n \in E\) can be written as a product of two or more elements in \(E\), we say that \(n\) is \(E\) -composite; otherwise, we say that \(n\) is \(E\) -prime. As examples, 4 is \(E\) -composite and 6 is \(E\) -prime. Find a necessary and sufficient condition for an integer to be an \(E\) -prime. Prove your statement.

Use the Euclidean algorithm to find the greatest common divisor of each pair of integers. $$ 67942,4209 $$

Show that \(\operatorname{gcd}(n, \phi)=1,\) and find the inverse s of \(n\) modulo \(\phi\) satisfying \(0
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