Is there a limit to the number of times I can format a hard disk?
I tried to find this information on Wikipedia, but I didn't find an answer.
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With the exception of CDs, DVDs and Blu-Ray discs (collectively called "optical media"), formatting is not a special action, and is fundamentally the same as any other disk operation. Formatting a storage device (whether it's a hard drive (HDD), solid state disk (SSD), or flash drive) just involves regular old reads and writes to the disk.
The only matters of concern are:
All this writing of new data (which will vary in quantity depending on what type of format you performed and what you're going to do after you format it) can cause wear on SSDs, and to a lesser degree, the mechanical parts of HDDs. The amount of wear is proportional to the amount of data that is being read/written, with SSDs largely unaffected by reads, but HDDs being affected roughly the same by reads and writes.
I'm not going to delve into the topic of disk endurance and how certain quantities and frequencies of reads and writes affect the endurance (wear level) of different types of disks. This is a very complex topic that is completely independent from the subject of disk formatting.
Just know that the operation of reinstalling Windows on a hard disk is basically doing the same thing to your disk as copying several gigabytes of movies or pictures or music. Just the act of using a computer involves very frequent disk reads and writes.
The only difference is that formatting a disk and then using it often incurs a fairly large amount of reads and writes compared to what a typical user might do in a day.
Analogy: if you normally drive 8 km to work every day in your car, and then take a holiday trip of 200 km, this is fundamentally the same action -- you're just driving further. Formatting causes more wear on your disk, just like driving further causes more wear on your car.
If you want to know how reading and writing data impacts the endurance of your particular type of disk, you can either ask a new question, or search for existing questions (or use Google) to find this information.
An SSD's service life is rated in total write cycles and is very precise. A Traditional spinning hard drive's service life is rated in hours of runtime, and can only be estimated in a very general way. It's important to note that HDDs and SSDs wear differently.
The flash memory cells in an SSD have a hard limit to the number of erase/rewrite cycles they can tolerate before they die. This number is fixed and predictable, so it is very easy to determine how "healthy" an SSD is. SSDs employ wear leveling algorithms in their firmware to spread out that wear and extend the life of the device.
Spinning hard drives are a different story. The individual magnetic bits can be flipped back and forth an infinite number of times. Furthermore, the mechanical parts (head armature and spindle motor) are controlled by magnets as well so they do not physically make contact against each other. Hard drives wear from heat, humidity, vibration, shock, microscopic defects in the manufacturing process, etc.
Doing a full, long format of an SSD WILL degrade the drive and is unnecessary most of the time.
Doing a full, long format of a spinning hard drive will not degrade the hard drive per se. The wear comes from the heads thrashing about, not by flipping bits back and forth.
Long story short, no, there is no effective limit to the number of times you can format a traditional spinning hard drive -- at least nothing beyond simply using it for the same length of time.
The mechanism that wears out in hard disk drives is that which moves the heads in and out, "seeking". Over time they lose their calibration. Note that this takes a LONG time these days (not once the case). I bet that if you were curious enough, you could find the number of seeks that a HD maker expects from their drives. A quick format, which is mostly on track 0, causes minimal motion and minimal wear.