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Introduction to Decimal to Binary
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Today, we're going to learn about converting decimal numbers into binary. Can anyone tell me why this conversion is important in digital electronics?
Because computers operate using binary data.
Exactly! Computers and digital systems work with binary. Now, letβs break down how we convert an integer from decimal to binary. What do you think is the first step?
We should divide the number by 2 and keep track of the remainder.
Correct! We divide the number and remember the remainders until the quotient is 0. Letβs do a quick example together. What is the binary equivalent of 13?
If I divide 13 by 2, I get 6 with a remainder of 1.
Great! So, what's next?
Then we keep dividing 6 by 2 until we get 0.
Thatβs right! After finding all the remainders, we write them in reverse order. We'll recap this in the end!
Coffee Break
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So we have learned how to convert the integer part of a decimal number to binary. What do you think is next?
The fractional part?
Right! For the fractional part, we multiply by 2. Can someone explain how this works?
We multiply the fractional number by 2 and note the integer part until we get 0 or reach a specific number of digits.
Exactly! This method is often called the double-dabble method. Let's apply this to 0.375 together.
Yeah! 0.375 times 2 equals 0.75. So the carry is 0.
Perfect! Keep going with that process.
Putting It All Together
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Now we've converted both integer and fractional parts. Whatβs the final result for 13.375?
The integer part is 1101 and the fractional part is .011!
Excellent! So when we combine those, the binary representation is 1101.011. Now, who can remind us why learning this is important?
It helps us understand how computers process numbers!
Exactly! Great job, everyone! Always remember, practice makes perfect!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section details the process of decimal-to-binary conversion utilizing the double-dabble method. It involves successive division for the integer part and multiplication for the fractional part, providing a systematic approach to representation in binary format.
Detailed
Decimal-to-Binary Conversion
In this section, we explore how to convert decimal numbers into binary format, an essential skill in digital electronics. Conversion involves dealing with both the integer and fractional parts separately, using two distinct procedures:
- Integer Part: To find the binary equivalent, we divide the integer by 2, recording the remainders. The process continues until the quotient reaches zero. The binary representation is then formed by reading these remainders in reverse order.
- Fractional Part: The conversion for the fractional part is carried out by multiplying the fractional part by 2, keeping track of the integer part (carry). This process continues until the result is zero or until we've achieved the desired number of binary digits. It can be referred to as the double-dabble method, a handy way to ensure accurate representation.
Example
For instance, consider converting the decimal number 13.375 to binary:
1. Integer part (13):
- Divide by 2 and keep track of remainders:
- 13 Γ· 2 = 6 (remainder 1)
- 6 Γ· 2 = 3 (remainder 0)
- 3 Γ· 2 = 1 (remainder 1)
- 1 Γ· 2 = 0 (remainder 1)
- Reading the remainders in reverse gives us 1101.
- Fractional part (0.375):
- Multiply by 2 and take note of the carries:
- 0.375 Γ 2 = 0.75 (carry 0)
- 0.75 Γ 2 = 1.5 (carry 1)
- 0.5 Γ 2 = 1.0 (carry 1)
- The binary equivalent here is .011.
Thus, the complete binary representation of 13.375 is 1101.011.
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Introduction to Decimal-to-Binary Conversion
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As outlined earlier, the integer and fractional parts are worked on separately.
Detailed Explanation
In the process of converting a decimal number to binary, we handle two parts separately: the integer part and the fractional part. This means that we will apply different methods for each part to find their binary equivalents.
Examples & Analogies
Think of converting a grocery bill (like breaking it into total cost and tax) and converting both parts individually to manage your budget.
Converting the Integer Part
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For the integer part, the binary equivalent can be found by successively dividing the integer part of the number by 2 and recording the remainders until the quotient becomes β0β. The remainders written in reverse order constitute the binary equivalent.
Detailed Explanation
To convert the integer part of a decimal number into binary, follow these steps: divide the number by 2. Write down the remainder from this division. Repeat this process with the quotient until you reach a quotient of zero. Once done, write all the remainders in reverse order - from the last remainder you got to the first.
Examples & Analogies
Imagine you have a stack of coins, and you're sorting them into piles of two. Each time you sort, you take a pile away and keep track of any leftover coin (remainder). The piles represent the binary digits, and when you combine them all together in reverse order, you see how much you have in binary form.
Converting the Fractional Part
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For the fractional part, it is found by successively multiplying the fractional part of the decimal number by 2 and recording the carry until the result of multiplication is β0β. The carry sequence written in forward order constitutes the binary equivalent of the fractional part of the decimal number.
Detailed Explanation
When converting the fractional part, you multiply the number by 2. Record the whole number part (the carry) that results from this multiplication. Then take the new fractional part (the part after the decimal) and repeat the multiplication by 2. Keep doing this until you reach a result of zero, or for a set number of bits. Write down the carries you recorded in the order they were created to form the binary equivalent.
Examples & Analogies
Consider cooking where you double a recipe each time. Each time you double, you take note of how much food you've added to the dish (the carry) and what remains (the new fractional part). You keep doubling until you reach your target amount, thus keeping track of all additions.
Example of Decimal-to-Binary Conversion
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The process can be best illustrated with the help of an example. Example 1.3 We will find the binary equivalent of (13.375). Solution: β’ The integer part = 13. Divisor Dividend Remainder 2 13 β 2 6 1 2 3 0 2 1 1 β 0 1 β’ The binary equivalent of (13) is therefore (1101). β’ The fractional part = .375 0.375 Γ 2 = 0.75 with a carry of 0 0.75 Γ 2 = 0.5 with a carry of 1 0.5 Γ 2 = 0 with a carry of 1 β’ The binary equivalent of (0.375) = (.011) β’ Therefore, the binary equivalent of (13.375) = (1101.011).
Detailed Explanation
In this example, first we convert the integer part '13' to binary. We divide by 2 repeatedly, noting down remainders, leading us to find that 13 in decimal is 1101 in binary. For the fractional part '0.375', we multiply it by 2, keeping track of any whole numbers we get. For example, 0.375 multiplied by 2 gives 0.75 (carry 0), and multiplying 0.75 by 2 gives 1.5 (carry 1), wrapping up the fractional conversion. When put together, 13.375 becomes 1101.011 in binary.
Examples & Analogies
Think of a digital clock (for hours) and a timer (for minutes) together showing how long you have baked cookies. The hours show the whole number (the integer part), and the minutes show fractional segments of the hour that represent a smaller portion. Converting both tells you how far youβve gone in your baking journey.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Conversion Process: The systematic method involves handling both integer and fractional components separately.
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Remainder Method: For the integer part conversion, we record remainders after dividing by 2.
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Multiplication Method: The fractional part is generated by multiplying by 2 and recording the carries.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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For the decimal number 13, the binary equivalent is determined by successive division, yielding 1101.
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In converting 0.375, multiplying by 2 repeatedly gives .011 as the binary representation.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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When dealing with decimal, do not fear, just divide by two and the answer will appear!
π Fascinating Stories
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Imagine a bakery where you divide the dough (decimal) to find how many pieces are in each (binary), and the leftover is noted!
π§ Other Memory Gems
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DIME for Decimal to Integer to Multiply for the fractional part!
π― Super Acronyms
BID
- Binary Integer Divide for the integer part.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Binary Equivalent
Definition:
The representation of a decimal number in the binary numeral system.
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Term: DoubleDabble Method
Definition:
A systematic process for converting decimal fractions into binary.
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Term: Quotient
Definition:
The result obtained from dividing one number by another.
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Term: Remainder
Definition:
The amount left over when a number cannot be divided evenly.
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Term: Fractional Part
Definition:
The part of a decimal number that consists of numbers to the right of the decimal point.
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Term: Integer Part
Definition:
The part of a decimal number that consists of its whole number component.