Math Is Fun Forum

I completely agree that Einstein's early work was his best. Einstein had other gift's as well. I often wonder how different the world might be today had Einstein had accepted the presidency of Israel when it was offered to him. In my mind that was his greatest mistake, and a tremendous loss for us today.

"God does not play dice."

I think that things often appear random because we do not understand the underlying patterns, or we do not have the ability to make sufficient calculations.

That would place a lot of importance on us. We live on an average planet that orbits an average sun in an average galaxy . . .

The mathematical models of the laws of the universe are being created to more detail as our ability to make more precise observations increases. Newton wasn't wrong. He just left his work incomplete. Einstein added precision to Newton's work that was unneeded and unobservable in Newton's time. One day someone will find a model that is more precise than relativity. It is the models that change, not the laws.

The digit seven occurs once every ten digits. That is to say that between 1 and ten, you will find seven once.

100 is 10 * 10, so we know seven will occur at least 10 times per 100 consecutive numbers. However, it occurs ten times from 70 to 79. We've already accounted for 77, though, so we only need to count the other 9. Thus, the digit seven occurs in 19 of the numbers between 1 and 100, inclusive.

The same holds true for the numbers between 101 and 200, between 201 and 300, etc. There are 9 blocks of 100 numbers between 100 and 1000, so, since 19 * 9 = 171, there are at least 171 numbers between 100 and 1000 that contain the digit seven.

However, all the numbers between 700 and 799 contain the digit seven. However, we are now only interested in the ones we haven't counted. Since we have already counted 19 of them, we only need to add another 81.

171 + 81 = 252

Make sense?

the first second of the eleventh minute of the eleventh hour of the day, actually. The tenth hour is over at 10:00:00.

The Devil was one of God's perfect creations, also. If the devil is not doing exactly as God intended, then God was not perfect in his creation, which would mean God is not perfect or all powerful.

Interesting. How did you obtain that, if you don't mind me asking?

There. I fixed it for ya! :D:D:P

Aha, the floor function.

So we have:

       prove that lim of f(x) as x approaches 2 , exists and is not equal to f(2)

When x = 2:

When 1 < x < 2:

When 2 < x < 3:

However, I have been under the impression that the limit of f(x) as x approaches n is equal to f(n) if f(n) is defined. That seems to not be the case in this example, but I'm not positive. Perhaps a more knowledgeable poster will clarify.

Edit to add: This particular f(x) is not a continuous function, so we cannot assume that the limit of f(x) as x approaches n is equal to f(n).

I also found the wording somewhat confusing.

I interpreted "Fifty minutes ago, if it was four time as many minutes past three o'clock" as 50 minutes ago it was 4*50 minutes past three o'clock, which meant the "current" time was 250 minutes past three o'clock, or ten minutes after seven o'clock. By that reasoning, it would be six o'clock in ten hours and fifty minutes, or 650 minutes.

I want to emphasize that I am not claiming you are incorrect; I just did not understand your interpretation of the problem.

Mathsy beat me again. I only got a 126.

Cosines of opposite corners should be additive inverses, so the cosines of two pair of opposite angles should sum to zero also.