Kara: That number is your grandfather's serial number which is what was used before the military started using social security numbers. That number normally is preceeded by two letters which indicated the serviceman's branch of service. For the enlisted in the Army, those two letters would be: RA if the person was part of the Regular Army, ER if he was part of the reserves (ER=Enlisted Reserves if I remember correctly) and NG if he was in the National Guard. If you were transferred from one component to another, then the number would remain the same and the letters would change. Since your grandfather was Regular Army, then his full service number was RA43064382. If he had gotten out of the Regular Army and transferred to the National Guard, his full service number would change to NG43064382.
For officers in the Army, their service number if they were officers in the Reserve, would start with the letter O and be followed by 7 numbers, such as O5436732. If they were officers in the Regular Army, their service number would start with two letters, OF and they would have 6 or more numbers following those letters, thus, their service number would be something like OF234667.
Hope this helps.
Tom