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MySQL 101 : Numeric type attributes and maximum value

I would like to make a quick MySQL 101 about numeric types in MySQL and especially explain the difference between the numeric type attributes and the maximum value that a numeric type can store.

Someone asked me yesterday why when he inserts a value in a table, the value 127 appears instead of the value 160 that he’s just inserted.
Ok, let me explain, consider this table :

What happens if you want to insert this row in the table :

sql > insert into product values (160, 554, "iPad mini");
Query OK, 1 row affected, 1 warning (0.01 sec)
sql > show warnings;
| Level   | Code | Message                                     |
| Warning | 1264 | Out of range value for column 'id' at row 1 | …
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