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.
$$-4 \times 10^{50}$$
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?
$\endgroup$ 115 Answers
$\begingroup$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.
$\endgroup$ 6 $\begingroup$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 $-(a\cdot b)$.
$\endgroup$ 1 $\begingroup$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.
$\endgroup$ 2 $\begingroup$Scientific notation is a way to shorten a value/measure of a thing in reality. For example: Mass of an atom of Oxygen is $0.0000000000000000000002677931574$ gram. It is a positive number, but extremely small. To be easier for writing this value, scientists created a kind of notation, called scientific notation by using exponent/power of $10$. Therefore, the above number can be written such as $2.677931514 x 10^{-22}$ ($10$ power to negative $22$) So, do not get confused with a negative number of it. Scientific notation is NEVER used for a negative number. An extremely small negative number can borrow the scientific notation to shorten its writing, but the concept of "scientific notation" is just for a positive number of the value of the real things in the REALITY!
$\endgroup$ 1 $\begingroup$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
$\endgroup$ 2
