Percentage points of the normal distribution ~ statistics

I am meant to find the value of a in P(Z
So I looked in the table of normal distribution function for 0.9990 in P(Z
However, the book requires you to do 1-0.9990 = 0.0010 and using the percentage points table, leading to the (correct) answer of 3.0902

Why was my first method wrong? I don't see the difference.

Your method is perfectly correct. 3.10 should get you full credit.The readings at the
end of the table are not very accurate since the tail probabilities are very small over
a wide range of z.
I suspect that the percentage table is calculated more accurately.

As you perhaps know, the x-axis for the Normal-distribution is either expressed by the mean: the mean +/- 1,2,3 sigma, or: z = (x-my)/sigma. You are using the second x-axis, and then the numbers 1 to 3 (and -1 to -3) are showed on both sides of the y-axis.

How you look up the probabilities depends on the table-structure. Usually, it starts with probability 0,5 at z=0. But then - instead of reading out the probability of 0,999, you can read out the value of (-0,001) at the other side of the z-axis and then get the answer: 1 - table-value.

You have perhaps used to different tables, and the number of significant digits can have been different (x-axis) in these two tables.