Converting Decimal Fractions to Base 5
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Questions and Answers

What is the first step in converting the fractional part of a decimal number to another base?

  • Group the deducted integers.
  • Deduct the integer part from the product.
  • Multiply the fraction by the base. (correct)
  • Repeat the process until the fraction is zero.

What should be done with the integer part obtained during the conversion process?

  • It should be kept aside for later use. (correct)
  • It should be added to the new value.
  • It should be converted back to decimal.
  • It should be ignored completely.

When do you stop the conversion process of the fractional part?

  • When the integer part becomes too large.
  • When a rounded number is achieved.
  • When the base has been reached.
  • When the remaining fraction is zero or enough digits are obtained. (correct)

What is the final step after obtaining several integer parts during conversion?

<p>Group the deducted integers from top to bottom. (A)</p> Signup and view all the answers

If you want to convert a decimal fraction to base 5, how will the first integer be determined?

<p>By multiplying the fraction by 5. (C)</p> Signup and view all the answers

What is the first integer result when converting 0.430410 to base 5?

<p>2 (A)</p> Signup and view all the answers

How many multiplication steps are required before reaching a fractional part of zero when converting 0.430410 to base 5?

<p>Three (C)</p> Signup and view all the answers

What is the final base 5 representation of the decimal fraction 0.430410?

<p>0.20345 (A)</p> Signup and view all the answers

What would be the result of the second multiplication step (0.1520 × 5) during the conversion process?

<p>0.7600 (B)</p> Signup and view all the answers

What base is being used in the conversion of 0.430410?

<p>Base 5 (A)</p> Signup and view all the answers

What is the second bit in the binary representation of 0.3437510 when converted to binary?

<p>1 (A)</p> Signup and view all the answers

Which operation is performed to convert the fractional part of a decimal number to binary?

<p>Multiply by 2 (B)</p> Signup and view all the answers

In converting 0.3437510 to binary, what fractional value is derived after the fourth multiplication?

<p>0.75000 (A)</p> Signup and view all the answers

What is the correct binary representation of 0.3437510 up to 4 bits to the right of the binary point?

<p>0.01012 (B)</p> Signup and view all the answers

After the first multiplication of 0.34375 by 2, what integer part is produced?

<p>0 (C)</p> Signup and view all the answers

What is the first step when converting a number from any base to decimal?

<p>Multiply each digit by the base raised to the power of its position (B)</p> Signup and view all the answers

When converting from base n to decimal, what power should the least significant bit (LSB) have?

<p>0 (D)</p> Signup and view all the answers

What should you do after multiplying each digit by the base raised to its power?

<p>Add all the intermediate results together (D)</p> Signup and view all the answers

How are the positions of digits after the decimal point treated when converting to decimal?

<p>They have negative powers starting from -1 (D)</p> Signup and view all the answers

If you have a number in base 4 represented as '312', what is the value of the digit '3' in decimal?

<p>12 (B)</p> Signup and view all the answers

What method is used to convert the fractional part from decimal to another base?

<p>Multiplication by the base and taking the integer part (D)</p> Signup and view all the answers

Why do we use multiplication instead of the remainder method for fractional parts?

<p>Because fractions are expressed in negative powers of the base (A)</p> Signup and view all the answers

In a base conversion, what is the significance of the integer part obtained from the multiplication process?

<p>It represents the new fractional part in the target base (B)</p> Signup and view all the answers

Which of the following best describes how fractions in decimal are expressed when converted to another base?

<p>Using only negative powers (C)</p> Signup and view all the answers

Which of the following is an example of a fraction in base 10?

<p>2390167.1208 (D)</p> Signup and view all the answers

What is the primary relationship between octal and binary systems?

<p>Each octal digit corresponds to a group of 3 binary digits. (D)</p> Signup and view all the answers

What is the grouping size for converting binary to hexadecimal?

<p>4 binary digits per hexadecimal digit. (C)</p> Signup and view all the answers

Which of the following statements is true regarding binary conversions?

<p>Binary numbers can be converted directly to octal and hexadecimal. (D)</p> Signup and view all the answers

Which base is equivalent to hexadecimal in terms of power-of-two notation?

<p>Base 16 (D)</p> Signup and view all the answers

If 11101100 is split for hexadecimal conversion, which grouping is correct?

<p>1110 1100 (B)</p> Signup and view all the answers

What is the decimal equivalent of the binary number 10.11?

<p>2.75 (D)</p> Signup and view all the answers

What is the value of the letter 'A' in hexadecimal when converting to decimal?

<p>10 (C)</p> Signup and view all the answers

When converting the hexadecimal number 1A.5 to decimal, what is the contribution of the digit '5'?

<p>0.3125 (B)</p> Signup and view all the answers

If you convert the hexadecimal number 2F.3 to decimal, what is the contribution of the digit '2'?

<p>16 (D)</p> Signup and view all the answers

For the binary number 110.101, what is the total final value when converted to decimal?

<p>6.5 (C)</p> Signup and view all the answers

What is the first step in converting the Hex number 351B to Octal?

<p>Convert Hex to Binary (B)</p> Signup and view all the answers

After converting Hex to Binary, how are the binary bits grouped for the conversion to Octal?

<p>In sets of three bits (A)</p> Signup and view all the answers

Which binary value represents the Hex digit 'B' during conversion?

<p>1010 (B)</p> Signup and view all the answers

What is the Octal equivalent of the Hex number 351B after completing the conversion?

<p>32433 (A)</p> Signup and view all the answers

Why is it useful to convert through Binary when changing from Hex to Octal?

<p>It ensures accuracy in converting between bases. (C)</p> Signup and view all the answers

What is the primary reason for converting a number to base 10 when changing bases?

<p>It simplifies the conversion process. (B)</p> Signup and view all the answers

Which scenario is an exception to the rule of converting to base 10 for base conversion?

<p>Converting between bases that are powers of two. (B)</p> Signup and view all the answers

When converting a number from Base x to Base y, which step is considered as the first in the conversion process?

<p>Convert the number to base 10. (A)</p> Signup and view all the answers

If converting a number directly between two bases is sometimes possible, what should be considered first?

<p>The relationship between the two bases. (C)</p> Signup and view all the answers

What type of bases might allow for faster conversions without using base 10 as an intermediary?

<p>Bases that are powers of two. (B)</p> Signup and view all the answers

What is the first step in converting binary to octal?

<p>Make groups of 3 bits from right to left for the integer part (C)</p> Signup and view all the answers

When completing an octet during conversion, where do you add zeros for the integer part?

<p>On the left side (C)</p> Signup and view all the answers

What should you do with the digits after converting each octet to octal?

<p>Concatenate the digits without any operations (A)</p> Signup and view all the answers

How are the bits grouped for the fractional part when converting from binary to octal?

<p>From left to right (B)</p> Signup and view all the answers

If the binary number is 1011011.101, how should you group it for octal conversion?

<p>10 110 11, 110 000 (B)</p> Signup and view all the answers

Flashcards

Fractional Part Conversion

The process of transforming the fractional part of a decimal number into an equivalent value in a different base (e.g., binary, octal, hexadecimal).

Base n

The base of the target number system (e.g., 2 for binary, 8 for octal, 16 for hexadecimal).

Step 1: Multiplication

Multiply the fractional part of the decimal number by the base n.

Step 2: Deduction

Subtract the integer part of the product obtained from the multiplication step.

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Step 3: Repetition

Use the remaining fractional part from the deduction step and repeat the process from step 1 until either the fraction becomes zero or enough digits are obtained.

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Converting Decimal Fractions to Other Bases

The process of converting a decimal fraction to a fraction in another base by repeatedly multiplying the fractional part by the desired base and taking the integer part of the result.

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Decimal Fractions and Negative Powers

Decimal fractions are represented using negative powers of the base. For example, 0.125 in base 10 is equivalent to 1/10^1 + 2/10^2 + 5/10^3.

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Place Value in Different Bases

In base 10, we use powers of 10: 10^0, 10^1, 10^2, etc. In base 2, for example, we use powers of 2: 2^0, 2^1, 2^2, etc.

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Conversion Algorithm

To convert a decimal fraction to another base, we repeatedly multiply the fractional part by the desired base. The integer part of the product becomes the next digit of the fraction in the new base, and the remaining fractional part is carried to the next iteration.

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Repeating Fractions

The process of converting a decimal fraction may result in a repeating fraction in the new base. For example, 1/3 in base 10 is a repeating fraction in base 2 (0.010101...).

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Decimal to Base Conversion

The process of changing a number from decimal (base-10) to another base system.

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Destination Base

The base to which you want to convert the decimal number. For example, base 5, base 2 (binary), or base 16 (hexadecimal).

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Repeated Multiplication Method

Repeatedly multiplying the fractional part of a decimal number by the destination base.

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Integer Part

The integer part of the result after each multiplication in the repeated multiplication method. These integers form the digits of the number in the new base.

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Zero Fractional Part

When the fractional part of the result becomes zero after repeated multiplication. This indicates the conversion is complete.

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Fractional Part

The part of a decimal number that appears after the decimal point.

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Decimal to Binary Conversion (Fractional Part)

The repetitive multiplication of the fractional part of a decimal number by the new base, while capturing the integer part of each result.

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Bits to the Right of the Binary Point

The number of digits present in the fractional part of a binary number.

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Conversion Stopping Criteria

The process of terminating the conversion when a predetermined number of bits to the right of the binary point are obtained, aiming to approximate the original decimal value.

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Base Conversion to Decimal

The process of converting a number from any base (e.g., base n) to the decimal system.

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Least Significant Bit (LSB)

The digit furthest to the right; it has a positional value of 0 (n^0).

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Base Conversion to Decimal Formula

A simple way to remember how to convert any base to decimal. Multiplying each digit by the base raised to its position and adding the result gives the decimal representation.

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Place Value in Base Conversion

In a base-n system, each digit's value is multiplied by n raised to the power of its position, starting from 0 for the rightmost digit.

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Hexadecimal to Decimal Conversion

Convert a hexadecimal number to decimal by multiplying each hex digit by 16 raised to the power of its position. For letters, use their decimal equivalents (A=10, B=11, etc.).

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Binary to Decimal Conversion

Convert a binary number to decimal by multiplying each binary digit by 2 raised to the power of its position, starting from the rightmost digit as 0.

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Intermediate Base Conversion

Converting a number from one base to another by first converting it to base 10 (decimal) and then to the desired base.

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Base Conversion

The process of transforming a decimal number (base 10) into a number in another base system (e.g., binary, octal, hexadecimal).

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Converting Decimal Fractions

Converting a decimal fraction to another base by repeatedly multiplying the fractional part by the desired base and taking the integer part of the product.

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Direct Conversion (Powers of Two)

Direct conversion between bases that are powers of two (e.g., binary, octal, hexadecimal) is often faster and simpler than using base 10 as an intermediate step.

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What is an octet?

A group of 3 binary digits, each representing a single octal digit.

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What is a hextet?

A group of 4 binary digits, each representing a single hexadecimal digit.

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How do we convert from binary to octal or hexadecimal?

Directly grouping binary bits into groups of 3 to obtain octal digits, or groups of 4 to obtain hexadecimal digits.

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What is place value in the binary system?

In binary, each position represents a power of 2. For example, 1011 in binary is equivalent to 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 1 * 2^0.

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Why are power-of-two bases important?

Binary, octal, and hexadecimal bases are important in computer science for their efficient representation and manipulation of data.

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Binary to Octal Grouping

Grouping binary digits into sets of three, starting from the rightmost digit for the integer part and the leftmost digit for the fractional part, to convert them into octal.

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Completing Octets

Adding leading zeros to the left of the integer part and trailing zeros to the right of the fractional part to complete the last group of 3 bits.

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Octal Digit Conversion

Converting each group of 3 bits into its corresponding octal digit.

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Concatenating Octal Digits

Joining the octal digits obtained from each group without any addition to form the final octal representation.

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Octal Octet

A set of three binary digits representing a single octal digit.

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Digit-by-Digit Conversion

Breaking down a number in one base into its equivalent representation in another base using the place values of each digit. This is done by grouping digits and converting them one by one.

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Hex to Binary Conversion

Representing each digit of a hexadecimal number with a group of four binary bits. Each binary group corresponds to a single hexadecimal digit.

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Binary to Octal Conversion

Grouping the binary bits in sets of three, where each group represents an octal digit. This utilizes the fact that 8 is equal to 2 raised to the power of 3.

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Binary as an Intermediate Step

Converting a number from one base to another by first converting it to binary and then converting the binary representation to the desired base.

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