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

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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