1.4 Scientific Notation

Writing Numbers in Standard or Scientific Notation Form

When we say that a number is in scientific notation, we are specifying the form in which that number is written. That form begins with an integer with an absolute value between 1 and 9, then perhaps followed by the decimal point and then some more digits. This is then multiplied by 10 raised to some power. When the number only has one non-zero digit, the scientific notation form is the digit multiplied by 10 raised to an exponent. When the number has more than one non-zero digit, the scientific notation form is a single digit, followed by a decimal, which is then followed by the remaining digits, which is then multiplied by 10 to a power.

The following numbers are written in scientific notation:

$$1.45\times {10}^3$$

$$-8.345\times {10}^{-4}$$

$$3\times {10}^2$$

$$3.14159\times {10}^0$$

The following numbers are not written in scientific notation:

$1.45$ because it isn't multiplied by 10 raised to a power

$-50.053\times {10}^7$ because the absolute value of −50.053 is not at least 1 and less than 10

$41.7\times {10}^9$ because 41.7 is not at least 1 and less than 10

$0.036\times {10}^{-3}$ because 0.036 is not at least one and less than 10

Some numbers are so large or so small that it is impractical to write them out fully. Avogadro’s number is important in chemistry. It represents the number of units in 1 mole of any substance. The substance may be electrons, atoms, molecules, or something else. Written out, the number is: 602,214,076,000,000,000,000,000. Another example of a number that is impractical to write out fully is the length of a light wave. The wavelength of the color blue is about 0.000000450 to 0.000000495 meters. Such numbers are awkward to work with, and so scientific notation is often used. We need to discuss how to convert numbers into scientific notation, and also out of scientific notation.

Recall that multiplying a number by 10 adds a 0 to the end of the number or moves the decimal one place to the right, as in $43\times 10=430$ or $3.89\times 10=38.9$. And if you multiply by 100, it adds two zeros to the end of the number or moves the decimal two places to the right, and so on. For example, $38\times \mathrm{100,000}=\mathrm{3,800,000}$ and $32.998\times \mathrm{10,000}=\mathrm{329,980}$. Multiplying a number by 1 followed by some number of zeros just adds that many zeros to the end of the number or moves the decimal place that many places to the right. Numbers written as 1 followed by some zeros are just powers of 10, as in ${10}^1=10$, ${10}^2=100$, ${10}^3=\mathrm{1,000}$, etc. Generally, ${10}^n=1\underset{n}{\underbrace{\mathrm{0...0}}}$.

We can use this to write very large numbers. For instance, Avogadro’s number is 602,214,076,000,000,000,000,000, which can be written as $6.02214076\times {10}^{23}$. The multiplication moves the decimal 23 places to the right.

Similarly, when we divide by 10, we move the decimal one place to the left, as in $\frac{46.7}{10}=4.67$. If we divide by 100, we move the decimal two places to the left, as in $\frac{3.456}{100}=0.03456$. In general, when you divide a number by a 1 followed by $n$ zeros, you move the decimal $n$ places to the left, as in $\frac{\mathrm{8,244.902}}{\mathrm{1,000,000}}=0.008244902$. This denominator could be written as ${10}^6$. If we use that in the expression and allow for negative exponents, rewrite the number as $$\frac{\mathrm{8,244.902}}{\mathrm{1,000,000}}=\frac{\mathrm{8,244.902}}{{10}^6}=\mathrm{8,244.902}\times {10}^{-6}$$. With this, we can write division by a 1 followed by $n$ zeros as multiplication by 10 raised to $‒n$.

Using that information, we can demonstrate how to convert from a number in standard form into scientific notation form.

Case 1: The number is a single-digit integer.

In this case, the scientific notation form of the number is $digit\times {10}^0$.

Case 2: The absolute value of the number is less than 1.

Follow the process below.

• Step 1: Count the number of zeros between the decimal and the first non-zero digit. Label this $n$.
• Step 2: Starting with the first non-zero digit of the number, write the digits. If the number was negative, include the negative sign.
• Step 3: If there is more than one digit, place the decimal after the first digit from Step 2.
• Step 4: Multiply the number from Step 3 by ${10}^{-(n+1)}$.

Case 3: The absolute value of the number is 10 or larger.

Follow the process below.

• Step 1: Count the number of digits that are to the left of the decimal point. Label this $n$.
• Step 2: Write the digits of the number without the decimal place, if one was present. If the number was negative, include the negative sign.
• Step 3: If there is more than one digit, place the decimal point after the first digit.
• Step 4: Multiply the number from Step 3 by ${10}^{n-1}$.

When we write numbers in scientific notation form, we can manipulate the representation of the number by moving the decimal around, and making an appropriate change to the exponent of the 10. For instance, let’s look at $145.8141\times {10}^8$. If we wanted to move the decimal one place to the left, we’d have to increase the power of 10, as shown here: $$145.8141\times {10}^8=14.58141\times {10}^9$$. Since we moved the decimal one to the left, we balance that with moving the exponent up by one. Similarly, if we move the decimal one place to the right, we have to balance that by moving the exponent one to the left, or subtracting one from the exponent, as shown here: $$145.8141\times {10}^8=1458.141\times {10}^7$$. Generally, for a number in the form $number\times {10}^n$:

• If you move the decimal to the left by $k$ digits, you increase the exponent by $k$.
• If you move the decimal to the right by $k$ digits, you decrease the exponent by $k$ digits.

Converting Numbers from Scientific Notation to Standard Form

In the previous section, converting a number from standard form to scientific notation was explored. Now, we explore converting from scientific notation back into standard form. Doing so involves moving the decimal according to the power of the 10. The decimal is moved a number of steps equal to the exponent of the 10. As demonstrated previously, when the exponent of the 10 is negative, the decimal is moved to the left and when the exponent of the 10 is positive, the decimal is moved to the right.