Scientific notation and negative numbers

My daughter is learning scientific notation in school, and her textbook says something to the effect of this:

Scientific notation is a method of writing numbers as the product of two factors where the first factor is a number greater than or equal to 1 but less than 10 and the second factor is a power of 10.

The teacher is taking this to mean that you cannot express a negative number in scientific notation. So that e.g.

$-4 * 10^{50}$

would not be valid scientific notation because -4 is less than 1.

Is there such a view of scientific notation? It certainly doesn't jive with my memory (or wikipedia), or is that description just deficient, and should better read:

Scientific notation is a method of writing numbers as the product of two factors where the first factor is a number whose absolute value is greater than or equal to 1 but less than 10 and the second factor is a power of 10.

And if it is a legitimate view, how do you express negative numbers in scientific notation?

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Teacher is wrong. – mjqxxxx Dec 7 '12 at 4:53
Sounds like crap to me. – copper.hat Dec 7 '12 at 4:56
Your reformulation is still inadequate, as it can't express $0$. – Chris Eagle Dec 7 '12 at 8:14
Maybe the teacher can watch this: youtube.com/watch?v=0Elky_dDak0 – Amzoti Dec 7 '12 at 9:34
@ChrisEagle: when do you need to write $0$ in the scientific notation ? And with what exponent ? – Yves Daoust Sep 25 '15 at 13:51

You are right. The textbook and teacher are wrong.

Scientific notation is where numbers are written in the form $a × 10^b$ where $a ∈ ℝ$ and $b ∈ ℤ$.

Normalized scientific notation also stipulates that $1 ≤ |a| < 10$.

$\therefore \;\; -4 × 10^{50}$ is correct normalized scientific notation, as common sense would dictate.

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Everyone is right, there is no incompatibility ! – Yves Daoust Sep 25 '15 at 16:08
@YvesDaoust: According to the textbook definition “and the second factor is a power of 10”, $7×1000000$ is scientific notation. This is incorrect. “The teacher [says] you cannot express a negative number in scientific notation.” This is also wrong: $−1×10^0$ expresses $-1$ in scientific notation. – Zaz Sep 25 '15 at 17:50
I wasn't clear. $1\times 10^0$ is scientific notation; according to the teacher's definition, $-1\times 10^0$ is not scientific notation. But it is a valid number anyway: the negative of $1\times 10^0$. These two points of view are quite reconciliable. – Yves Daoust Sep 25 '15 at 19:07
@YvesDaoust: Sorry if I seemed short. Your point that $(-a)b = -(ab)$ is a good one, but from my point of view it only makes the statement “you cannot express a negative number in scientific notation” even more meaningless. Ahhh... after considering the question “Can you express a negative number in decimal?” I think I see your point perfectly. Technically the answer is no, for that you'd need a negative base or $-$. In any case, I think what the teacher said is unhelpful, certainly if she didn't also explain that you could stick a minus sign in front of it to make it negative. – Zaz Sep 26 '15 at 0:30
My point is that what the teacher said is harmless. – Yves Daoust Sep 26 '15 at 8:12

There is no problem having a negative number shown in scientific notation, computers and calculators have been doing this for ages. The quote from the textbook shouldn't be taken too literally as they were probably thinking of the magnitude of the number and forgot about this trivia.

In any case, the problem collapses if you read $-4\cdot10^{50}$ as $-(4\cdot10^{50})$. The expression inside the parentheses is a valid number written in the scientific notation, and there is no reason to forbid taking the opposite. (Though the whole expression itself cannot be called a "number written in the scientific notation" if you apply the definition as it stands.) Remember that by usual rules of precedence, $-a\cdot b$ is understood as $-(a\cdot b)$.

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As regards myself, I would not even object against notations like $4000\cdot10^{47}$ or $0.004\cdot10^{53}$ which still make sense (denormalized scientific notation). – Yves Daoust Sep 25 '15 at 12:58

By now you may have noticed that, if put to a vote, "Yes, you can use negative numbers too." would win but not by a unanimous vote. There was a time, when people used slide rulers, that scientific notation was needed to perform calculations. It still is now, but not nearly as much. Now it's just a simple way to represent really large and really small numbers. Once your daughter gets out of that class, I don't think anyone will complain if she expresses negative numbers using scientific notation.

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So you are saying that for slide rulers there were no negative numbers? – Yishai Sep 27 '15 at 1:16
@Yishai. My God. Am I really that old? No, they were not necessary. Slide rules were made, mostly, to multiply numbers. You had to work out the powers of 10 and the sign of the product. – Steven Gregory Sep 27 '15 at 2:41

Scientific notation is a method of writing numbers as the product of two factors where the first factor is a number WITH AN ABSOLUTE VALUE greater than or equal to 1 but less than 10 and the second factor is a power of 10.

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You would write the problem as: $|-4| \times 10^{50}$, or $4\times 10^{50}$. Your teacher is right, you cannot express a scientific notation as a negative number. It has to be a number greater than or equal to one, but less than 10. The absolute value would be taken, therefore, giving you $4 \times 10^{50}$. I just did one like this in college Chemistry. Thanks! MB

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Sorry, I'm not following what you are trying to say. – Yishai Jan 29 '15 at 22:33
To be more specific, I'm not seeing how you are expressing (or distinguishing) a negative. – Yishai Jan 29 '15 at 22:40