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Chapter 5: Problem 56

Does 30470 represent a number in binary? in octal? in decimal? in hexadecimal?

Short Answer

Expert verified
No, 30470 does not represent a binary or octal number. Yes, it does represent a decimal and a hexadecimal number.

Step by step solution

01

Analyze in Binary system

In binary, only digits 0 and 1 are valid. Examine the digits in the number 30470. The number 30470 includes digits other than 0 and 1, thus it does not represent a number in binary system.
02

Analyze in Octal system

In octal system, only digits from 0 to 7 are valid. Examine the digits in the number 30470. The number 30470 includes the digit 8 which is not valid in octal system. Hence, it does not represent a number in octal system.
03

Analyze in Decimal system

In decimal system, digits from 0 to 9 are valid. Examine the digits in the number 30470. All the digits in 30470 are in this range. Hence, 30470 does represent a number in decimal system.
04

Analyze in Hexadecimal system

In hexadecimal system, digits from 0 to 9 and letters from A to F are valid. Examine the digits in the number 30470. All the digits in 30470 fall in this range. Thus, the number 30470 does represent a number in hexadecimal system.

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Key Concepts

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

Binary System
The Binary System is a number system that uses only two symbols: 0 and 1. It is widely used in computers and digital systems because they operate using an on (1) and off (0) state. Every binary number is a sequence composed solely of these two digits. These binary digits (often called bits) are the foundation of all computer data processing. Each bit in a binary number represents a power of 2, starting from the right with 2 to the power of 0. In summary, binary is the most straightforward number system for electronic devices to handle due to its simplicity and efficiency.
Octal System
The Octal System is another form of number system that uses eight distinct symbols, from 0 to 7. This base-8 system was often used in early computing systems and is still sometimes applied in modern computing for compact representation of binary numbers. Each position in an octal number represents a power of 8. For instance, the rightmost digit is the "ones" place (or 8 to the power of 0), the next is the eights place (8 to the power of 1), and so on. Since each octal digit represents three binary digits, it provides a more concise way to handle large binary numbers.
Decimal System
The Decimal System, also known as base-10, is the most common number system used by humans in day-to-day life. It employs ten symbols: 0 through 9. Our everyday numeric counting and mathematical operations are accomplished using this system. The decimal system is centered around powers of 10. Each position in a decimal number represents a power of 10, making it intuitive due to its alignment with our natural counting. Decimal numbers are everywhere and are the standard in financial and engineering calculations worldwide.
Hexadecimal System
The Hexadecimal System, or base-16 system, is a number system that uses sixteen symbols. It includes the digits 0 through 9 and the letters A through F, where A to F represent the values 10 to 15. Hexadecimal numbers are a compact representation of binary numbers, often used in computing and programming. Each hexadecimal digit corresponds to four binary digits, allowing for a concise and easy-to-read representation of binary data. This system is particularly beneficial for expressing long binary sequences in a shorter form, which can save space and make programming easier to manage.

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Most popular questions from this chapter

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. Is \(8 E\) -prime or \(E\) -composite?

Add the hexadecimal numbers. $$ 82054+\text { AEFA3 } $$

Prove that a base \(b\) integer \(m\) has \(\left\lfloor 1+\log _{b} m\right\rfloor\) digits.

Give an example of consecutive primes \(p_{1}=2, p_{2}, \ldots, p_{n}\) where $$ p_{1} p_{2} \cdots p_{n}+1 $$ is not prime.

Show that if \(p\) is a prime number, \(a\) and \(b\) are positive integers, and \(p \mid a b,\) then \(p \mid a\) or \(p \mid b\).
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