Math Problem Statement

First, convert the decimal number 199 to a quaternary (base-4) number. Base-4 uses the digits 0, 1, 2, and 3. Then convert this base-4 number to a binary number as follows. Starting from the right most digit, take each digit of the base-4 number and convert it to a 2- digit binary number. For example, the base-4 number 3021 can be converted to a binary number by converting each base-4 digit to 2-digit binary. We can use the following table to convert base-4 number to a 2-digit binary number. Base 4 2-digit binary 0 00 1 01 2 10 3 11 Using the above table, if we convert each of the digit 3, 0, 2, 1 to a 2-digit binary, we will get the following: 3 = 11, 0 = 00, 2 = 10, 1 = 01. Combining them, we get (3021) base-4 = (11001001) base-2 Compare the binary number you got for 199 with the result you got in problem number 1(a). Do the same with the other numbers 97, 227, 150, 214 from question 1. First convert these to quaternary number, then to a binary number and compare your results with what you got in problem 1 above. What do you notice?

Solution

Here are the conversions from decimal to quaternary (base-4) and then to binary (base-2) for the given numbers:

  1. 199

    • Base-4: 301343013_4
    • Binary: 11000111211000111_2

  2. 97

    • Base-4: 120141201_4
    • Binary: 01100001201100001_2

  3. 227

    • Base-4: 320343203_4
    • Binary: 11100011211100011_2

  4. 150

    • Base-4: 211242112_4
    • Binary: 10010110210010110_2

  5. 214

    • Base-4: 311243112_4
    • Binary: 11010110211010110_2

Observation:

When converting from decimal to base-4 and then to binary, the patterns in the resulting binary numbers should be consistent with the method you described (i.e., converting each base-4 digit to a 2-digit binary). The final binary number is a direct representation of this.

Do you have any specific comparison with earlier results that you need further explanation on?

Would you like more details on:

  1. Comparing these binary numbers with earlier results?
  2. How the conversion process works step-by-step?
  3. Verifying base-4 and binary number accuracy?
  4. Practical applications of base conversion?
  5. Understanding patterns in base-4 or binary systems?

Tip: When converting between number systems, breaking it into smaller steps (like converting to intermediate bases) can simplify complex calculations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Number Systems
Base Conversions
Binary Numbers
Quaternary Numbers

Formulas

Decimal to Base-4 conversion
Base-4 to Binary conversion using a table: Base 4 → 2-digit Binary (0 → 00, 1 → 01, 2 → 10, 3 → 11)

Theorems

Positional Number System

Suitable Grade Level

Grades 8-10

Related Recommendation

Step-by-Step Conversion Exercises: Decimal, Binary, Octal, and Hexadecimal

Convert 175 from Base-10 to Base-4 Number

Decimal to Binary Conversion: 1234, 5670, 2321

Converting 27 from Base 10 to Binary (Base 2)

Converting Between Decimal, Binary, Octal, and Hexadecimal Number Systems