By the discovery of numbers I refer to Leonardo da Pisa, the so-called Fibonacci. To me Roman Numerals were not numbers. They were a counting device and could do nothing more.
Numbers had been used in the Arabic world. But Fibonacci recognized the importance of that magical number that made everything possible, the ZERO
The zero made it possible to create multiplication tables and addition and subtraction and division and everything related to calculation. I can't imagine what the world would be like if we still used Roman Numerals! I have this as my house number MCCCLXXI. There are people that think the McClicksi's live here!
But with the advent of this other great Leonardo, we have a lot of new ideas, such as significant figures.
Most of my students at the University level had no real idea of the meaning of the phrase, significant figures. They were told something in the fourth grade about the number of decimal places and try to use it as the only definition!
There are two basic situations, addition/subtraction and multiplication/divison. They are quite different from each other.
Addition and Subtraction
Here the consideration is the number of decimal places. The answer cannot have more decimal places than the piece of data that has the least. For instance, if I add 4.2 to 0.645 , the answer is 4.8, not 4.845. The limiting piece of information is 4.2.
Multiplication and Division
However, in multiplication and division you have to do a little more work. You have to calculate the "parts per thousand" accuracy of the various pieces of data. For instance if you say 10, you are really saying it is between 9.5 and 10.5. This means it is accurate to one part in ten! To carry it further, if you say 0.0045, this is accurate to 1 part in 45 or about 2%. Once you learn to do this, you have it licked.
You will probably have to use scratch paper and think about this a little to believe it. Of course, that is why so few people do it right! I used to take only one point off of a twenty point question when a student abused this by more than one decimal place. If it was abused by more than about two places I would take off more!
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The other half of the calculation is to accept the idea that the error in the result is the sum of the errors of the data going into it. If you have something like x = (1.245)(0.00462)/126.074 the calculator answer is 456,232.05. But what is the legitimate answer. The number 1.235 is accurate to 1 in 1245 or 0.810 parts per thousand. The number 0.00462 is accurate to one part in 462 or 1.73 parts per thousand. And 126.074 is accurate to 1 part in 126,074 or 0.00793185 parts per thousand. Now we add these errors together, whether it is multiplication or division, to get the accuracy of the answer. First of all we have to round all the errors to two decimal places, getting 0.80 + 1.73 +0.001 to get 2.54 parts per thousand or 0.254%. That makes us round the answer to456***.* or 4.56 x 10 to the fifth power.