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For a homework assignment on significant digits, I have to find the number of significant digits in the number $0.40 \times 10^3$. I know that in scientific notation, the number of significant digits is equal to the number of digits in the stem. However, I've also learned that for a number to be in scientific notation, it must have a stem between $1$ and $10$. What is the rule for this situation? Is the number of Significant digits $2$ or $3$?

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Most of the answers to your questions can be found in this wikipedia article. You are right if you say:

I've also learned that for a number to be in scientific notation, it must have a stem between 1 and 10.

This is the convention in scientific notation. The number format of $0.40 \times 10^3$ is called exponential notation (at least it was called that way in a book about the programming language Fortran I'm reading at the moment): Here the number has a stem between 0 and 1 followed by a power of 10. This is quite common in computer science. As to the significant digits: This wikipedia article (especially the paragraph "Scientific notation") contains the information you need. According to its information I would say that the number $0.40 \times 10^3$ has $2$ significant digits.

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"I've also learned that for a number to be in scientific notation, it must have a stem between $1$ and $10$".

We have: $0.40\times10^{3}$

$\hspace{13 mm}=4.0\times10^{2}$

In this form, the number fits the requirement of the stem being between $1$ and $10$.

So, we can see that the number has two significant figures.

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