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Exploring Binary to Decimal Conversion
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Today we will start with binary-to-decimal conversion. Can anyone tell me what a binary number looks like?
Isn't it just a sequence of 0s and 1s? Like, maybe (1010)?
Exactly! Now, to convert the binary number to decimal, we multiply each bit by its place value. For instance, in the binary number (1001), what are the place values starting from the right?
Well, they would be 2^0, 2^1, 2^2, and 2^3, which are 1, 2, 4, and 8!
Correct! Now, what do you get when you calculate the decimal equivalent of (1001)?
So it's 1 times 8 plus 0 times 4 plus 0 times 2 plus 1 times 1, which equals 9!
Perfect! So, (1001) in binary is indeed 9 in decimal. Remember the acronym BDM: Binary to Decimal Method!
BDM, got it! What about the fractional part, though?
Great question! The fractional part requires us to multiply by negative powers of 2. For example, for the binary (0.0101), we would evaluate 0 x 2^-1 + 1 x 2^-2 + 0 x 2^-3 + 1 x 2^-4!
Oh, so that leads to 0.3125 when calculated!
Exactly! In total, (1001.0101) equals 9.3125 in decimal. Always treat integer and fractional parts separately!
Let's summarize: When converting binary to decimal, remember to multiply each binary digit by its corresponding power of 2. Now, how do we convert octal to decimal?
Understanding Octal to Decimal Conversion
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Who can tell me what an octal number is composed of?
Octal numbers are made of digits from 0 to 7, right?
Correct! To convert an octal number to decimal, we will also multiply each digit by its respective place value, like in our previous example. For instance, letβs consider the octal number (137).
That gives us 1 x 8^2 + 3 x 8^1 + 7 x 8^0, which equals 64 + 24 + 7!
Exactly! So, what's the total?
That's 95!
Good! Now let's look at the fractional part in that octal number. Can anyone describe how to calculate it?
We multiply the fractional digits by decreasing powers of 8. For example, for .21, it would be 2 x 8^-1 + 1 x 8^-2.
Exactly! What is the result of that calculation?
That sums to 0.265 when added together!
Yes! Therefore, (137.21) in octal translates to 95.265 in decimal. Remember our phrase: Octal to Decimal with OPD - Octal Place Value Distribution!
Got it! What's next?
Hexadecimal to Decimal Conversion
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Now, let's talk about hexadecimal numbers. Who can remind us what digits are allowed in this base 16 system?
We can use 0 to 9 and A to F, right? So A is 10 and F is 15?
That's absolutely correct! Let's use the hexadecimal number (1E0). How do we convert that to decimal?
That would mean 1 x 16^2 + 14 x 16^1 + 0 x 16^0! A is 10, so E is 14!
Excellent calculation! What is the combined total?
So that's 256 + 224 + 0, which totals 480.
Yes! Now, what about converting the fractional part, .2A?
For that, we'd calculate 2 x 16^-1 + 10 x 16^-2.
That's right! What decimal value does that lead to?
That gives us 0.164 when added!
Exactly! So, the final decimal equivalent of (1E0.2A) would be 480.164. Always remember: Hex to Decimal with HFD - Hexadecimal Forward Distribution!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, the methods to convert numbers from binary, octal, and hexadecimal systems into their decimal equivalents are described in detail. The process involves calculating the values of integer and fractional parts separately based on their respective base values.
Detailed
Finding the Decimal Equivalent
This section focuses on converting numbers from binary, octal, and hexadecimal systems into their decimal equivalents. The decimal representation is computed by summing the products of the digits and their respective place values. It emphasizes treating the integer and fractional parts of the numbers separately for conversion.
Key Concepts
- Binary to Decimal Conversion: For a binary number like (1001.0101), the decimal equivalent is calculated by evaluating the integer part and the fractional part separately. The integer part achieves a value of 9, while the fractional part contributes an additional 0.3125, combining to yield a decimal output of 9.3125.
- Octal to Decimal Conversion: The section provides insight into converting an octal number such as (137.21) into its decimal form, yielding a total of 95.265.
- Hexadecimal to Decimal Conversion: The conversion of a hexadecimal number like (1E0.2A) into decimal form (480.164) incorporates a similar methodology of calculating integer and fractional contributions based on their base values.
These foundational conversions across different number systems highlight the significance of understanding base systems in computing, which is crucial for digital electronics.
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Example of 2's Complement Decimal Conversion
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Find the decimal equivalent of the following binary numbers expressed in the 2's complement format:
(a) 00001110;
(b) 10001110.
Solution:
(a) The MSB bit is '0', which indicates a plus sign.
The magnitude bits are 0001110.
The decimal equivalent = 0 Γ 20 + 1 Γ 21 + 1 Γ 22 + 1 Γ 23 + 0 Γ 24 + 0 Γ 25 + 0 Γ 26 = 0 + 2 + 4 + 8 + 0 + 0 + 0 = 14.
Therefore, 00001110 represents +14.
(b) The MSB bit is '1', which indicates a minus sign.
The magnitude bits are therefore given by the 2βs complement of 0001110, i.e. 1110010.
The decimal equivalent = 0 Γ 20 + 1 Γ 21 + 0 Γ 22 + 0 Γ 23 + 1 Γ 24 + 1 Γ 25 + 1 Γ 26 = 0 + 2 + 0 + 0 + 16 + 32 + 64 = 114.
Therefore, 10001110 represents -114.
Detailed Explanation
The 2βs complement format allows the representation of negative numbers in binary. To convert from 2's complement format, first check the most significant bit (MSB). If it's '0', the number is positive. Calculate the decimal equivalent similarly to what we've done previously. If the MSB is '1', the number is negative, calculate its positive counterpart by finding the 2's complement first (flip the bits and add one) and convert it just like a positive binary before putting the negative sign.
Examples & Analogies
Consider how speedometers display speed. For example, if you're driving at 60 mph and then reverse at -30 mph, you can use the same measurement system. If the indicator shows a '1' at the far left, you're going forward; if it shows '0,' you're moving in reverse (like a negative in 2's complement). The approach for converting these speeds (or numbers) ensures you correctly interpret both positive and negative movements.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Binary to Decimal Conversion: For a binary number like (1001.0101), the decimal equivalent is calculated by evaluating the integer part and the fractional part separately. The integer part achieves a value of 9, while the fractional part contributes an additional 0.3125, combining to yield a decimal output of 9.3125.
-
Octal to Decimal Conversion: The section provides insight into converting an octal number such as (137.21) into its decimal form, yielding a total of 95.265.
-
Hexadecimal to Decimal Conversion: The conversion of a hexadecimal number like (1E0.2A) into decimal form (480.164) incorporates a similar methodology of calculating integer and fractional contributions based on their base values.
-
These foundational conversions across different number systems highlight the significance of understanding base systems in computing, which is crucial for digital electronics.
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 binary number (1011), the decimal equivalent can be calculated as 1Γ2^3 + 0Γ2^2 + 1Γ2^1 + 1Γ2^0 = 11.
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The octal number (75) converts into decimal as 7Γ8^1 + 5Γ8^0 = 61.
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The hexadecimal number (2F) converts to decimal, where 2Γ16^1 + 15Γ16^0 (F=15) = 47.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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In binary, just bits do play, To decimal theyβll find their way. Multiply and add, donβt stray!
π Fascinating Stories
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Once there was a binary tree, with branches named 0 and 1. Each branch led to a decimal number where they combined and had fun. The key was the place value, as they danced to the count of two!
π§ Other Memory Gems
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Remember BDM: Binary to Decimal Method, OPD: Octal Place Distribution, HFD: Hexadecimal Forward Distribution.
π― Super Acronyms
Remember B.O.H.
- Binary
- Octal
- Hexadecimal for quick conversion strategies.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Binary Number
Definition:
A number represented in base 2, consisting only of the digits 0 and 1.
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Term: Octal Number
Definition:
A number represented in base 8, using the digits from 0 to 7.
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Term: Hexadecimal Number
Definition:
A number represented in base 16, using the digits from 0 to 9 and letters A to F.
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Term: Place Value
Definition:
The value of a digit based on its position in a number.
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Term: Decimal Equivalent
Definition:
The representation of a number in base 10.