When I round 6396 to 3 significant figures, I get 6400. I was double checking my work and used scientific notation for 6400, which is 6.4*10³, but when I counted the significant figures in scientific notation, I only get 2, which are 6 and 4, but there were supposed to be 3. Anyone can help to explain?
The problem is the rounding. Since you round up, it results in a loss of significant digits. If you are going to use scientific notation, and you want three sf's, then use 6.40*10^3
As explained here:
http://www.astro.yale.edu/astro120/SigFig.pdf
When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant:190 miles may be 2 or 3 significant figures, 50,600 calories may be 3, 4, or 5 significant figures. The potential ambiguity in the rule can be avoided by the use of scientific notation. For example, depending on whether 3, 4, or 5 significant figures is correct, we could write 50,6000 calories as:
5.06×10^4calories (3 significant figures)
5.060×10^4calories (4 significant figures), or
5.0600×10^4calories (5 significant figures).
When using scientific notation, the significant figures are determined by the nonzero digits. In the case of 6.4 * 10³, there are indeed three significant figures (6, 4, and the decimal point after 4).
However, when you round a number to a certain number of significant figures, the final result should reflect the precision of the original number. In this case, 6396 has four significant figures, so when rounded to three significant figures, it becomes 6400. In scientific notation, the equivalent representation of 6400 would be 6.4 * 10³, with only two significant figures (6 and 4).
To recap:
- The number 6.4 * 10³ in scientific notation has three significant figures.
- The rounded number 6400, when expressed in scientific notation, becomes 6.4 * 10³, which has two significant figures.
Sure! Let's break down the concept of significant figures and how they apply to scientific notation.
Significant figures are used to indicate the precision or accuracy of a measurement or number. The rules for determining significant figures are:
1. Non-zero digits are always significant.
2. Any zeros between non-zero digits are significant.
3. Leading zeros (zeros before the first non-zero digit) are not significant.
4. Trailing zeros (zeros at the end of a number) are significant if there is a decimal point present.
Now, let's apply these rules to the number you mentioned, 6400.
When you round 6396 to 3 significant figures, it becomes 6400. The reason for this is because we are rounding to the nearest thousand, and since the last digit (9) is greater than 5, we round up.
Now, let's consider the scientific notation representation of 6400, which is 6.4 x 10³.
In scientific notation, the coefficient (6.4) represents the significant figures, while the power of 10 (³) represents the magnitude of the number.
To determine the number of significant figures in scientific notation, we only consider the coefficient. In this case, 6.4 has two significant figures (6 and 4), as you correctly pointed out.
However, it's important to note that the power of 10 (³) does not contribute to the significant figures. It simply tells us that the decimal point needs to be moved three places to the right to obtain the full number.
So, to summarize, when you round 6396 to 3 significant figures, you get 6400, which has three significant figures. However, when representing 6400 in scientific notation, the coefficient (6.4) has only two significant figures.