"I would consider leading wi..."


by Eric Rogstad Jun 17 2016

When you learned that we use decimal notation to record numbers, you may have wondered: If we used a different number system, say a number system with 12 symbols instead of 10, how would that change the cost of writing numbers down? From the fact that the length of a written number grows logarithmically with the magnitude of the number and the above equation, we can see that, no matter how large a number is, its base 10 representation differs in length from its base 12 representation only by a factor of $~$\\log\_{10}(12) \\approx 1.08$~$\. Similarly, the binary representation of a number is always about $~$\\log\_2(10) \\approx 3.32$~$ times longer than its decimal representation\. Because there is only one logarithm function \(up to a multiplicative constant\), which number base you use to represent numbers only affects the size of the representation by a multiplicative constant\.

I would consider leading with this question, to motivate the post.