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Chapter 5: Problem 55

Does 1101010 represent a number in octal?

Short Answer

Expert verified
Yes, 1101010 is a number in octal.

Step by step solution

01

Understanding Octal Numbering System

The octal number system, or just octal, is a base-8 number system. This means it only uses eight symbols or digits to represent values, which are: 0, 1, 2, 3, 4, 5, 6, and 7.
02

Checking the Provided Number for Valid Digits

Having understood what the octal system entails, we now check the given number '1101010' to ensure it doesn鈥檛 contain any digit outside the range of 0-7. From left to right, the numbers are: 1-1-0-1-0-1-0.
03

Conclude

As we can see from step 2, all digits in '1101010' are either 1 or 0, which are valid in the octal system. Therefore, '1101010' is indeed a number in octal.

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Key Concepts

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

Base-8 Number System
The octal or base-8 number system is one of the various numbering systems used to express values. Unlike the decimal system, which most people are familiar with and is a base-10 system, octal modulates values using only 8 distinct symbols. This is because each place value in an octal number represents a power of 8.
For example, the first three place values in octal are 1s, 8s, and 64s.
Counting in octal goes from 0 to 7, then resets when reaching 8, similar to how the decimal system goes from 0 to 9 before resetting at 10. This feature makes octal particularly useful in fields like computing, where groupings of binary digits (bits) ideally convert to simpler forms.
Octal Digits
In the octal number system, we use a specific set of digits: 0, 1, 2, 3, 4, 5, 6, and 7. These are the only digits allowed in base-8 representation. It鈥檚 crucial to understand that each digit in an octal number has a specific place value that corresponds to a power of 8.
  • The rightmost digit represents the 80 place.
  • The next digit to the left represents the 81 place.
  • And so on as we move leftward.
Using these digits correctly, values in octal can represent surprisingly large numbers. For example, the octal sequence "123" is calculated as (1 脳 8^2 + 2 脳 8^1 + 3 脳 8^0 = 83 in decimal).
This conversion highlights the importance of knowing the correct digits and interpreting them accurately.
Digit Validity in Number Systems
When working with any number system, it's vital to confirm the validity of the digits used. Each system has a specific set of permitted digits. For instance, while a decimal system can have digits up to 9, the octal system restricts digits to between 0 and 7.
To check if a number belongs in the octal system, you must ensure all its digits fall within this range. For example, if given the number "1101010", we can verify that each digit鈥�1, 1, 0, 1, 0, 1, 0鈥� is within the valid set of octal digits.
  • If a digit like 8 or 9 appears, it cannot be part of the octal system because those digits exceed the allowed range.
  • Conducting this check helps us quickly assess and avoid errors in calculations.
This understanding reinforces our ability to work with different number systems effectively and accurately.

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Most popular questions from this chapter

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

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

Find the greatest common divisor of each pair of integers. $$ 15,15^{9} $$

In the octal (base 8) number system, to represent integers we use the symbols \(0,1,2,3,4,5,6,\) and \(7 .\) In representing an integer, reading from the right, the first symbol represents the number of I's, the next symbol the number of 8 's, the next symbol the number of \(8^{2}\) 's, and so on. In general, the symbol in position \(n\) (with the rightmost symbol being in position 0 ) represents the number of \(8^{n}\) 's. $$ 7711 $$

Use the following definition: \(A\) subset \(\left\\{a_{1}, \ldots, a_{n}\right\\}\) of \(\mathbf{Z}^{+}\) is \(a^{*}\) -set of size \(n\) if \(\left(a_{i}-a_{j}\right) \mid a_{i}\) for all \(i\) and \(j,\) where \(i \neq j, 1 \leq i \leq n,\) and \(1 \leq j \leq n .\) These exercises are due to Martin Gilchrist. Prove that for all \(n \geq 2,\) there exists a \(*\) -set of size \(n .\) Hint: Use induction on \(n .\) For the Basis Step, consider the set \\{1,2\\} For the Inductive Step, let \(b_{0}=\prod_{k=1}^{n} a_{k}\) and \(b_{i}=b_{0}+a_{i}\) for \(1 \leq i \leq n\).
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