"We should just make base-10 computers," Bob commented to his friend Randy. "Rather than old base-2, which they only made because they used on/off vacuum tubes. We have the technology today to build faster computers... think of a base-10 computer! It would be five times faster than a base-2 computer - think about it. What could take up a bazillion digits in base 2 - or should I say bits, of 0 or 1, would take up much less in base-10. Five times less digits, five times more efficient!"

"That's not necessarily true," mused Randy, scratching his chin. "We're defining Base Efficiency as this, where x is the number to be compared, y is the larger base and z is the smaller base: "

B_{ef}(x) in y|z = [digits of x in base z] / [digits of x in base y]

For example, 5 in base 10 takes up 1 digit ("5") whereas, in base 2, it takes up three digits ("101"). Thus:

B_{ef}(5) in 10|2 = 3/1 = 3

"That's not 5x the storage efficiency," Randy added. "That's only three. Look at 10, that's 4/2, or 2. One is only 1/1! No, I think the average overall efficiency of base 10 compared to base 2 is...."

What does Randy say next? Why? Also, what's the largest efficiency you can get in 10|2, and what number(s) give you that efficiency?

## Question

## unreality 1

"We should just make base-10 computers," Bob commented to his friend Randy. "Rather than old base-2, which they only made because they used on/off vacuum tubes. We have the technology today to build faster computers... think of a base-10 computer! It would be five times faster than a base-2 computer - think about it. What could take up a bazillion digits in base 2 - or should I say bits, of 0 or 1, would take up much less in base-10. Five times less digits, five times more efficient!"

"That's not necessarily true," mused Randy, scratching his chin. "We're defining Base Efficiency as this, where x is the number to be compared, y is the larger base and z is the smaller base: "

B

_{ef}(x) in y|z = [digits of x in base z] / [digits of x in base y]For example, 5 in base 10 takes up 1 digit ("5") whereas, in base 2, it takes up three digits ("101"). Thus:

B

_{ef}(5) in 10|2 = 3/1 = 3"That's not 5x the storage efficiency," Randy added. "That's only three. Look at 10, that's 4/2, or 2. One is only 1/1! No, I think the average overall efficiency of base 10 compared to base 2 is...."

What does Randy say next? Why? Also, what's the largest efficiency you can get in 10|2, and what number(s) give you that efficiency?Use spoilers and show your work

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