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Integer Part Conversion
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Today, we will learn how to convert decimal numbers to octal numbers, starting with the integer part. Can anyone tell me what we might need to divide by during this conversion?
Is it 8, since octal is base 8?
Correct, Student_1! We divide the integer part by 8 repeatedly. Let's do a quick practice. If I have the number 73, how would we start?
We divide 73 by 8, right?
That's exactly right! What do we get?
The quotient is 9 and the remainder is 1.
Great job! Now, what's the next step?
We divide 9 by 8 next.
Yes! What do we find?
The quotient is 1 and the remainder is 1 again.
Excellent! Keep dividing until you hit 0. Letβs quickly summarize the steps: Divide by 8, note the remainder, and repeat until the quotient is 0. How do we write our final answer?
We write the remainders in reverse order!
Perfect! So for 73, the octal equivalent is 111. Well done, everyone!
Fractional Part Conversion
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Now that we know how to convert the integer part, let's talk about the fractional part. Remember, we multiply by 8 here. Letβs take 0.75 as our example. What happens when we multiply it by 8?
We get 6, and the next decimal is 0.
Exactly! So, we take the integer part, which is what?
The integer part is 6.
Great! So we note that down. Next, what do we do with the 0 now?
We stop since there's no fraction left to multiply.
Correct! So the octal equivalent of 0.75 is .6. If we combine this with the integer conversion, what do we get for the final octal equivalent of 73.75?
We get 111.6 in octal!
Correct! Letβs recap that: For fractions, we multiply by 8 and take the integer part until we reach a stopping point. Fantastic work today, everyone!
Introduction & Overview
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Quick Overview
Standard
The section describes the decimal-to-octal conversion process, detailing the methods for converting both integer and fractional parts separately. The use of repeated division for integer conversion and repeated multiplication for fractional conversion is highlighted with an example.
Detailed
Decimal-to-Octal Conversion
The process of converting decimal numbers to octal numbers is systematic and follows a similar logic to decimal-to-binary conversion. This section covers the detailed methodology for handling both integer and fractional parts of decimal numbers.
Conversion Process
- Integer Conversion:
- To convert the integer part of a decimal number into octal, you divide the integer by 8 sequentially until the quotient becomes zero, while recording the remainders. These remainders, when read in reverse order, give the octal equivalent.
- Fractional Conversion:
- The fractional part is converted by multiplying it by 8 repeatedly. The integer part of the resulting product is noted in each step until the fractional part becomes zero or until a desired precision is achieved.
The section illustrates these processes with a practical example, reinforcing the concept with clear operational steps.
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Audio Book
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Introduction to the Conversion Process
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The process of decimal-to-octal conversion is similar to that of decimal-to-binary conversion. The progressive division in the case of the integer part and the progressive multiplication while working on the fractional part are by β8β which is the radix of the octal number system. Again, the integer and fractional parts of the decimal number are treated separately.
Detailed Explanation
Decimal-to-octal conversion involves treating the integer and fractional parts independently. For the integer part, we divide it by 8 repeatedly until we reach a quotient of 0 and note the remainders. For the fractional part, we multiply it by 8 until the fractional part becomes 0 or until we've obtained the desired number of digits.
Examples & Analogies
Think of the decimal-to-octal conversion like baking a cake. You need to deal with the cake batter (the integer part) and the frosting (the fractional part) separately. First, you prepare the batter by dividing it into portions (dividing by 8) and then you add frosting, layer by layer (multiplying by 8) until it looks just right.
Example of Decimal-to-Octal Conversion
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Example 1.4
We will find the octal equivalent of (73.75) (cid:2)
10
Solution
β’ The integer part = 73
Divisor Dividend Remainder
8 73 β
8 9 1
8 1 1
β 0 1
β’ The octal equivalent of (73) =(111)
10 8
β’ The fractional part = 0.75
β’ 0.75Γ8 = 0 with a carry of 6
β’ The octal equivalent of (0.75) =(.6)
10 8
β’ Therefore, the octal equivalent of (73.75) =(111.6)
10 8
Detailed Explanation
In the example of converting the decimal number 73.75 to octal, we first focus on the integer part 73. We divide it by 8, which gives us a quotient of 9 and a remainder of 1, then divide 9 by 8 which gives us a quotient of 1 and a remainder of 1, and finally dividing 1 by 8 gives a quotient of 0 and a remainder of 1. We read the remainders from bottom to top to get 111 in octal. For the fractional part 0.75, we multiply it by 8 which gives us 6 (the whole number part of the result), so we have .6 in octal. Combining these results, 73.75 in decimal equals 111.6 in octal.
Examples & Analogies
Imagine you have 73 candies. You can give each friend (represented by the divisor 8) a group of candies. You first see how many complete groups of 8 you can make (that's like your division) and whatever is left becomes the remainder. For the fractional part, think of it like trying to share a piece of cake; you keep splitting it until youβve given away enough to represent 6/8 of it, which is equivalent to 0.75.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Octal Conversion: The methods for converting decimal integers and fractions into octal format.
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Remainders: Used to find octal representation from integer division.
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Multiplication: Essential for deriving octal representation from decimal fractions.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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To convert 73 to octal: 73 / 8 = 9 remainder 1. 9 / 8 = 1 remainder 1. This gives 111 in octal.
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For 0.75, multiplying by 8 gives 6 and ending with no fractional part, the octal is .6.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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When converting to octal, divide by eight, Remainders stack in reverse, thatβs the fate.
π Fascinating Stories
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Imagine a race where numbers compete to reach octal land; The integers split by eight and gather their remainders, while fractions multiply by eight to find a friend.
π§ Other Memory Gems
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D.O.D for Decimal to Octal: Divide for the integer and Multiply for the decimal.
π― Super Acronyms
O.R.D for Octal Remainder Division.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Octal Number System
Definition:
A base-8 number system that uses digits from 0 to 7.
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Term: Decimal Number
Definition:
A base-10 number system using digits from 0 to 9.
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Term: Conversion
Definition:
The process of changing a number from one number system to another.