Some people omit zeros from numbers. For example, they treat all of the following as being equivalent:
In many computer programming languages, all of these are different. (Many languages default to base (5+5), but will use base (5+3) if there is a leading zero, or base (5+5+5+1) if there is a leading 0x.) Most non-programmers expect the first two examples to mean the same thing, and the last two examples to mean different things.
Conflating tens and hundreds. Some children, when learning to count to large numbers, try to do this...
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, …
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, …
Confusing partial derivatives with full derivatives.
Confusing demonstrating that something is true for a special case with proving that it is true for the general case. Of course, demonstrating that something is false for a special case does disprove the general case of a claim -- unless special exceptions are carved out. But it is not enough to prove that the general case is always false.
Thinking that units don't matter. If I travel 120 miles in 2 hours, what is my average speed? 1? 60? Um, no. Those answers should be 1 mile per minute, or 60 miles per hour.
Mis-squaring units. How large is a square mile? 1 mi² = 5280 ft² ? Um, no. 1 mi² = 1 * (5280 ft)² = 1 * 5280 ft * 5280 ft = 5280 * 5280 ft² = 27,878,400 ft².
Thinking that the calculator will tell them the correct number of significant figures. What is 1,000,000 - 100,000? 899,999.99999999? Um, no. The calculator had a rounding error. The precise answer is 900,000; rounding to the correct number of significant figures will produce either 900,000 or 1,000,000.
Rounding off intermediate values, instead of final answers. Computers are especially prone to this problem, which is why there are entire books on Numerical Methods. Intermediate values should be absurdly precise. Care needs to be exercised in interpreting them.
Throwing away the remainder. What is 10 / 9? 1? 1.1? 1.11? Um, no. It is 1 + 1/9, or 1.11… . That … is a number, and it is not zero. It might not be worth calculating right now, but if we multiply it by 9, we will get something useful. (This also makes it easier to understand why 10 = 9.99…; this … equals 9 * 0.01 / 9, which equals 0.01 .)