Scientific notation

Scientific notation is a way of writing numbers using powers of 10. The basic format is: \[ N \times 10^x \] where N is a decimal number between 1 and 10 (but never equal to 10), and x is an integer that indicates how many places the decimal point must move to return to the original number.

This method allows you to express extremely large or small numbers in a compact and easy-to-read form.

It’s especially useful when dealing with astronomical or subatomic data, as it saves you from having to write out long strings of digits.

Example: The average distance from the Earth to the Sun is about 150,000,000 km. Writing this number in full can be unwieldy, but in scientific notation, it becomes much clearer: $1.5 \times 10^8$ km. This makes the number easier to work with, both in writing and calculation.

How to convert a number to scientific notation
A practical example
Pros and cons of scientific notation

How to convert a number to scientific notation

Converting a number from standard form to scientific notation is simple and only takes a few steps. Let’s go through the process for both large and small numbers.

Numbers greater than 1
Move the decimal point to the left until it’s just after the first significant digit. The number of places moved will be the exponent of 10.
Example: The number 450,000 becomes $4.5 \times 10^5$, because the decimal point was moved 5 places.
Numbers less than 1
In this case, move the decimal point to the right until the first significant digit is in the ones place. The number of places moved will be a negative exponent.
Example: The number 0.00032 becomes $3.2 \times 10^{-4}$, because the decimal point was moved 4 places to the right.

Converting scientific notation back to standard form

The reverse process is equally straightforward. You simply move the decimal point in the coefficient either to the right or left, depending on the exponent of 10.

Positive exponent
If the exponent $x$ of $10^x$ is positive, move the decimal point to the right.
For example, $3.2 \times 10^6$ becomes 3,200,000.
Negative exponent
If the exponent $x$ of $10^x$ is negative, move the decimal point to the left.
For example, $4.56 \times 10^{-3}$ becomes 0.00456.

A practical example

Let’s look at a few real-world examples where scientific notation proves to be essential.

The mass of a proton is approximately $1.67 \times 10^{-27}$ kg.

Writing this number in full would be incredibly cumbersome.

$$ 0.00000000000000000000000000167 \ kg $$

It’s far more convenient to represent it using scientific notation.

$$ 1.67 \times 10^{-27} \ kg $$

Note: In this case, there are 26 zeros between the decimal point and the first significant digit (1). The exponent $-27$ indicates that the decimal point has been moved 27 places to the left from 1.67.

Example 2

The distance between Earth and the Andromeda galaxy is $2.365 \times 10^{22}$ meters.

Scientific notation makes it easier to work with such an enormous number.

In this case, you would move the decimal point 22 places to the right.

$$ 2,365,000,000,000,000,000,000 \ meters $$

Note: In astronomical terms, this distance is roughly 2.537 million light years ( $ 2.537 \times 10^6 $ ). Knowing that one light year is about $9.461 \times 10^{15}$ meters, multiplying by 2.537 million light years gives a value close to $2.365 \times 10^{22}$ meters. $$ (9.461 \times 10^{15} ) \times ( 2.537 \times 10^6 ) = 23.65 \times 10^{21} \ m = 2.365 \times 10^{22} \ m $$

Example 3

Planck’s constant, one of the fundamental constants in physics, is $6.626 \times 10^{-34}$ J·s.

Again, scientific notation makes it much easier to handle such a tiny number.

Pros and cons of scientific notation

Scientific notation is widely used in physics and chemistry for several reasons:

Minimizes errors. Working with very large or very small numbers in standard form can easily lead to mistakes. Scientific notation reduces this risk and makes calculations more precise.
Improved readability. The compact representation helps keep data clear and easy to understand.
Simplifies calculations. When multiplying or dividing numbers in scientific notation, you only need to add or subtract the exponents, making the process much easier.

Example 1: Let’s calculate an expression using numbers in scientific notation: $$ (2 \times 10^5) \times (3 \times 10^4) $$ To multiply in scientific notation, multiply the coefficients: $2 \times 3 = 6$, and add the exponents: $5 + 4 = 9$. The result is: $$ (2 \times 10^5) \times (3 \times 10^4) = (2 \times 3) \times 10^{5 + 4} = 6 \times 10^9 $$ This is much simpler than multiplying the original numbers 200,000 and 30,000, which would give you 6,000,000,000. $$ 200,000 \times 30,000 = 6,000,000,000 $$
Example 2: Now let’s look at a division problem with numbers in scientific notation: $$ \frac{4 \times 10^6}{2 \times 10^3} $$ As before, divide the coefficients: $\frac{4}{2} = 2$, and subtract the exponents: $6 - 3 = 3$. The final result is: $$ \frac{4 \times 10^6}{2 \times 10^3} = \left( \frac{4}{2} \right) \times 10^{6-3} = 2 \times 10^3 $$ Without scientific notation, you would need to divide 4,000,000 by 2,000, which is much harder to do mentally. In summary, scientific notation simplifies calculations by reducing them to basic operations on coefficients and exponents, making everything quicker and less error-prone.

However, there are a few drawbacks to consider:

Learning curve. Not everyone is immediately comfortable with scientific notation. It may take some time for those used to standard notation to adapt to the new format.
Limited use outside scientific fields. Scientific notation can be unintuitive for those unfamiliar with it, making it less suitable for communicating data to a broader audience.
More steps in non-elementary operations. For operations like addition or subtraction, you need to adjust the exponents to match, which adds an extra step compared to standard notation.
Approximation. Scientific notation often requires rounding numbers to a certain number of significant digits, which can introduce small inaccuracies in some situations.

Despite these drawbacks, scientific notation remains an indispensable tool in science and many technical fields, where its benefits far outweigh any initial challenges.

And so on.