True or false multiplying 0.001 is the same as dividing by 1,000?

True.
Example:
.001 times 1,000 is the same as moving the decimal of .001 to the right three times. ( =1)

1,000 divided by 1,000 is the same as moving the decimal of 1,000 to the right three times. (=1)

Both expressions have the same answer. So therefore, your statement is true.