How many times does the number 1 appear on a 24 hour digital clock?

1164 times in a day.

Starting from right to left:

We get the number 1 in the single-minutes place 6 times per hour (01, 11, 21, 31, 41, 51).

We get the number 1 in the ten-minutes place 10 times per hour (10, 11, 12, 13, 14, 15, 16, 17, 18, 19)

Together that's 16 times per hour that we get a 1 in the minutes half of the digital display, since there are 24 hours in the day that's 16*24 times per day which is 384.

We get the number 1 in the single-hours place for 3 out of 24 hours in a day (01, 11, 21) for a 24 hour clock.

We get the number 1 in the ten-hours place for 10 out of 24 hours in a day (10, 11, 12, 13, 14, 15, 16, 17, 18, 19) for a 24 hour clock.

So we see the number '1' in the hours field 13 times for every 24 hours that pass. Each hour is 60 minutes so the number 1 will display 13 * 60 times in a whole day in the hours field, which is 780.

Summing the 1's we see in the hours place in a day (780) and the numbers we see in the minutes place in a day (384) we get 780 + 384 = 1164