Binary and hexadecimal expansion

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The discussion focuses on demonstrating that the binary expansion of a positive integer can be derived from its hexadecimal expansion by converting each hexadecimal digit into a four-digit binary block. Participants express confidence in the validity of this concept but seek guidance on how to formally prove it. The mathematical expression for a hexadecimal number is provided, emphasizing its unique representation. The conversation highlights the need for a systematic approach to convert hexadecimal to binary. Ultimately, the task is to establish a clear method for this conversion process.
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Homework Statement


Show that the binary expansion of a positive integer can be obtained form its hexadecimal expansion by translating each hexadecimal digit into a block of four binary digits.


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The Attempt at a Solution


I know that this is true but I have no idea how to show it. Any pointers?
 
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If you have number given as ##a_n 16^n + a_{n-1} 16^{n-1} + ... + a_0##, can you express it as binary number? This expression is unique, so once you find some way to represent the number you are done.
 

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Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
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