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Chapter 4: Problem 64

Use repeated division by 2 to convert (by hand) the integers in \(63-65\) from base 10 to base 2 . $$ 98 $$

Short Answer

Expert verified
The base 2 representation of the given integer \(98\) is \(1100010\).

Step by step solution

01

Divide by 2 and record remainder and quotient

Start by dividing \(98\) by \(2\) and get the remainder. \[ \require{cancel} 98 \div 2 = 2 \times 49 + 0 \] Quotient: \(49\) Remainder: \(0\) Note down the remainder and use the quotient in the next step.
02

Repeat step 1 with the quotient

Divide the quotient (\(49\)) by \(2\) and get the remainder. \[ 49 \div 2 = 2 \times 24 + 1 \] Quotient: \(24\) Remainder: \(1\) Note down the remainder and use the quotient in the next step.
03

Repeat step 2 with the new quotient

Divide the quotient (\(24\)) by \(2\) and get the remainder. \[ 24 \div 2 = 2 \times 12 + 0 \] Quotient: \(12\) Remainder: \(0\) Note down the remainder and use the quotient in the next step.
04

Repeat step 3 with the new quotient

Divide the quotient (\(12\)) by \(2\) and get the remainder. \[ 12 \div 2 = 2 \times 6 + 0 \] Quotient: \(6\) Remainder: \(0\) Note down the remainder and use the quotient in the next step.
05

Repeat step 4 with the new quotient

Divide the quotient (\(6\)) by \(2\) and get the remainder. \[ 6 \div 2 = 2 \times 3 + 0 \] Quotient: \(3\) Remainder: \(0\) Note down the remainder and use the quotient in the next step.
06

Repeat step 5 with the new quotient

Divide the quotient (\(3\)) by \(2\) and get the remainder. \[ 3 \div 2 = 2 \times 1 + 1 \] Quotient: \(1\) Remainder: \(1\) Note down the remainder and use the quotient in the next step.
07

Last division

Notice that the quotient in the last step was 1. This is the stopping point for our repeated division. We write down the quotient as it is. Quotient: \(1\)
08

Get the base 2 representation

Arrange the remainders in reverse order, from the last step to the first step, to get the base 2 representation of the given number. \(1 \ 1 \ 0 \ 0 \ 0 \ 1 \ 0\) Now, we have converted the given integer, \(98\), to base 2 as: \(1100010\)

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Key Concepts

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

Repeated Division
To convert a decimal number to binary, we use an approach called repeated division. This method involves dividing the decimal number by 2 repeatedly and recording the remainders.
  • First, you start with your decimal number. For this example, let's consider the number 98.
  • Divide the number by 2.
  • Write down the remainder because this will be part of your binary result.
  • Take the quotient you obtained from the division and use it as the new number to divide again by 2.
  • Continue this process of dividing by 2, noting down remainders, and using the next quotient until the quotient becomes zero.
This cycle of repeated division is easy to follow and is how many binary conversions are tackled manually.
Decimal to Binary
Converting a decimal number to binary might seem complicated at first, but using repeated division makes it straightforward. Decimal numbers are what we commonly use in our daily counting systems, and they are based on 10. Binary numbers, on the other hand, are based on 2 and are crucial in computing because computers think in binary.
Here's how you visualize the conversion:
  • Start with a decimal number, for instance, 98.
  • Use the repeated division method to write down the sequence of remainders.
  • When the division results in a quotient of zero, you are ready to assemble your binary number.
  • Read the remainders backward, from the last to the first, to form the binary equivalent.
  • For 98, the remainders list is 0, 1, 0, 0, 0, 1, 1, which reverse to 1100010 when read backward.
Once you have done this a few times, converting decimal to binary will become second nature.
Remainders in Binary Conversion
Remainders play a pivotal role when using repeated division to convert a decimal number to binary. They essentially form the backbone of your binary result.
Here's the importance of remainders:
  • Each remainder can only be a 0 or 1, which directly corresponds to binary digits.
  • The remainders are recorded at each step of dividing the decimal number by 2.
  • These remainders create the binary digits from the least significant bit to the most significant bit.
In the example of converting 98 to binary, at each step of division by 2:
  • If a remainder is 0, it means the current quotient was even.
  • If a remainder is 1, it means the current quotient was odd and that 1 extra had to be subtracted to make it divisible by 2.
It's intriguing to see how these seemingly simple 0s and 1s combine to form any number in a system understood universally by computers.

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