Multiplying significant figures and decimal places

Question

One of my lecturers today said that multiplying a number leaves the number of significant figures to which that number is correct unchanged, however it doesn't leave the number of decimal places correct.

He gave the following example: $128\times 0.99687$.

According to him, if 0.99687 is correct to 4 dp then the product is not correct to 4dp. However if 0.99687 is correct to 4sf then the product is also correct to 4sf.

I am struggling to see why this is the case. I have tried thinking about how decimal places differ from significant figures but I never really learned this in detail, only the "mechanics" of how to work dp's and sf's out, so any help would be appreciated.

Answer 1

"Correct to 4 decimal places" means that the 4th digit after the decimal point remains the same after rounding. But if we multiply this by 1000 then that 4th digit is now the 1st one after the decimal point. So the number is no longer correct in the 4 decimal place - it is now only correct in the 1st decimal place.

Conversely, dividing by 1000 the number is now correct to the 7th decimal place!

This happens because multiplication (or division) changes the positions of all digits.

However, if as with 0.9969, there are only 4 significant digits in the number, then after multiplication (or division) by 1000 there will still be only 4 significant digits.