1.4.3: Scientific Notation

1.4.4 Learning Objectives

• Change numbers in standard form to scientific form
• Change numbers in scientific form to standard form
• Perform computations on numbers in scientific form

Standard Form to Scientific Form

Very large numbers such as \(43,000,000,000,000,000,000\) (the number of different possible configurations of Rubik’s cube) and very small numbers such as \(0.000000000000000000000340\) (the mass of the amino acid tryptophan) are extremely inconvenient to write and read. Such numbers can be expressed more conveniently by writing them as part of a power of 10.

To see how this is done, let us start with a somewhat smaller number such as \(2480\). Notice that

\(\begin{aligned}
\underbrace{2480}_{\text {Standard form }} &=248.0 \times 10^{1} \\
&=24.80 \times 10^{2} \\
&=\underbrace{2.480 \times 10^{3}}_{\text {Scientific form }}
\end{aligned}\)

Scientific Form

The last form is called the scientific form of the number. There is one nonzero digit to the left of the decimal point and the absolute value of the exponent on 10 records the number of places the original decimal point was moved to the left.

\(\begin{aligned}
0.00059 &=\dfrac{0.0059}{10}=\dfrac{0.0059}{10^{1}}=0.0059 \times 10^{-1} \\
&=\dfrac{0.059}{100}=\dfrac{0.059}{10^{2}}=0.059 \times 10^{-2} \\
&=\dfrac{0.59}{1000}=\dfrac{0.59}{10^{3}}=0.59 \times 10^{-3} \\
&=\dfrac{5.9}{10,000}=\dfrac{5.9}{10^{4}}=5.9 \times 10^{-4}
\end{aligned}\)

There is one nonzero digit to the left of the decimal point and the absolute value of the exponent of 10 records the number of places the original decimal point was moved to the right.

Scientific Notation

Numbers written in scientific form are also said to be written using scientific notation. In scientific notation, a number is written as the product of a number between and including 1 and 10 (1 is included,10 is not) and some power of 10.

Writing a Number in Scientific Notation

To write a number in scientific notation:

1. Move the decimal point so that there is one nonzero digit to its left.
2. Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.

Example 1

Write \(981\) in scientific notation.

The number \(981\) is actually \(981.\), and it is followed by a decimal point. In integers, the decimal point at the end is usually omitted.

\(981 = 981. = 9.81 \times 10^2\)

The decimal point is now two places to the left of its original position, and the power of \(10\) is \(2\).

Example 2

Write \(54.066\) in scientific notation.

\(54.066 = 5.4066 \times 10^1 = 5.4066 \times 10\)

The decimal point is one place to the left of its original position, and the power of \(10\) is \(1\)

Example 3

Write \(0.000000000004632\) in scientific notation.

\(0.000000000004632 = 4.632 \times 10^{-12}\)

The decimal point is twelve places to the right of its original position, and the power of \(10\) is \(−12\).

Example 4

Write \(0.027\) in scientific notation.

\(0.027 = 2.7 \times 10^{-2}\)

The decimal point is two places to the right of its original position, and the power of \(10\) is \(-2\)

Scientific Form to Standard Form

A number written in scientific notation can be converted to standard form by reversing the above process.

Converting from Scientific Notation

To convert a number written in scientific notation to a number in standard form, move the decimal point the number of places prescribed by the exponent on the 10.

Positive Exponent/Negative Exponent

Move the decimal point to the right when you have a positive exponent, and move the decimal point to the left when you have a negative exponent.

Example 5

Write \(4.673 \times 10^4\) in standard form.

The exponent of \(10\) is \(4\) so we must move the decimal point to the right \(4\) places (including 0's, if necessary).

\(4.6730 \times 10^4 = 46730\)

Example 6

Write \(2.9 \times 10^7\) in standard form.

The exponent of \(10\) is \(7\) so we must move the decimal point to the right \(7\) places (including 0's, if necessary).

\(2.9 \times 10^7 = 29000000\)

Example 7

Write \(1 \times 10^{27}\) in standard form.

The exponent of \(10\) is \(27\) so we must move the decimal point to the right \(27\) places (including 0's, if necessary).

\(1 \times 10^{27}= 1,000,000,000,000,000,000,000,000,000\)

Example 8

Write \(4.21 \times 10^{-5}\) in standard form.

The exponent of \(10\) is \(-5\) so we must move the decimal point to the left \(5\) places (including 0's, if necessary).

\(4.21 \times 10^{-5} = 0.0000421\)

Example 9

Write \(1.006 \times 10^{-18}\) in standard form.

The exponent of \(10\) is \(-18\) so we must move the decimal point to the left \(18\) places (including 0's, if necessary).

\(1.006 \times 10^{-18} = 0.000000000000000001006\)

Working with Numbers in Scientific Notation

Multiplying and Dividing Numbers Using Scientific Notation

There are many occasions (particularly in the sciences) when it is necessary to find the product or quotient of two numbers written in scientific notation. This is accomplished by using two of the basic rules of algebra.

For Multiplication:

Suppose we wish to find \((a \times 10^n)(b \times 10^m)\). Since the only operation is multiplication, we can use the commutative property of multiplication to rearrange the numbers.

\((a \times 10^n)(b \times 10^m) = (a \times b)(10^n \times 10^m)\)

Then, by the rules of exponents, \(10^n \times 10^m = 10^{n+m}\). Thus,

\((a \times 10^n)(b \times 10^m) = (a \times b) \times 10^{n+m}\)

The product of \((a \times b)\) may not be between \(1\) and \(10\), so \((a \times b) \times 10^{n+m}\) may not be in scientific form. The decimal point in \((a \times b)\) may have to be moved. There is an example of this in the exercises below.

Example 10

\(\begin{aligned}
\left(2 \times 10^{3}\right)\left(4 \times 10^{8}\right) &=(2 \times 4)\left(10^{3} \times 10^{8}\right) \\
&=8 \times 10^{3+8} \\
&=8 \times 10^{11}
\end{aligned}\)

Example 11

\(
\begin{aligned}
\left(5 \times 10^{17}\right)\left(8.1 \times 10^{-22}\right) &=(5 \times 8.1)\left(10^{17} \times 10^{-22}\right) \\
&=40.5 \times 10^{17-22} \\
&=40.5 \times 10^{-5}
\end{aligned}
\)

We need to move the decimal point one place to the left to put this number in scientific notation.
Thus, we must also change the exponent of \(10\).

\(
\begin{array}{l}
40.5 \times 10^{-5} \\
4.05 \times 10^{1} \times 10^{-5} \\
4.05 \times\left(10^{1} \times 10^{-5}\right) \\
4.05 \times\left(10^{1+(-5)}\right) \\
4.05 \times\left(10^{1-5}\right) \\
4.05 \times 10^{-4}
\end{array}
\)

Thus,
\(
\left(5 \times 10^{17}\right)\left(8.1 \times 10^{-22}\right)=4.05 \times 10^{-4}
\)

For Division:

Suppose we wish to find \(\dfrac{a \times 10^n}{b \times 10^m}\). Similar to multiplication we will separate the exponential terms from the constant terms using the multiplication property of fractions.

\(\dfrac{a \times 10^n}{b \times 10^m}= \dfrac{a}{b} \times \dfrac{10^n}{10^m}\)

Then, by the rules of exponents, \(10^n \div 10^m = 10^{n-m}\). Thus,

\(\dfrac{a \times 10^n}{b \times 10^m}= (a \div b) \times 10^{n-m}\)

Be attentive to any negative exponents when applying the subtraction rule of exponents. Remember to include the negative sign when performing the operation to get the proper exponent. Review the examples below for different scenarios.

The quotient of \((a \div b)\) may not be between \(1\) and \(10\), so \((a \div b) \times 10^{n-m}\) may not be in scientific form. The decimal point in \((a \div b)\) may have to be moved. There is an example of this in the exercises below.

Example 12

\(\begin{aligned}
\dfrac{9 \times 10^{7}}{5 \times 10^{4}} &= \\
&=\dfrac{9}{5} \times \dfrac{10^{7}}{10^{4}} \\
&=\dfrac{9}{5} \times 10^{7-4} \\
&=1.8 \times 10^{3}
\end{aligned}\)

Example 13

\(\begin{aligned}
\dfrac{8.7 \times 10^{-4}}{ 3.2\times 10^{3}} &= \\
&=\dfrac{8.7}{3.2} \times \dfrac{10^{-4}}{10^{3}} \\
&=\dfrac{8.7}{3.2} \times 10^{-4-3} \\
&=2.72 \times 10^{-7}
\end{aligned}\)

Example 13

\(\begin{aligned}
\dfrac{-5.3 \times 10^{-7}}{ 2\times 10^{-3}} &= \\
&=\dfrac{-5.3}{2} \times \dfrac{10^{-7}}{10^{-3}} \\
&=\dfrac{-5.3}{2} \times 10^{-7-(-3)} \\
&=\dfrac{-5.3}{2} \times 10^{-7+3} \\
&=-2.65 \times 10^{-4}
\end{aligned}\)

Example 14

\(\begin{aligned}
\dfrac{1.2 \times 10^{7}}{ 4.4\times 10^{9}} &= \\
&=\dfrac{1.2}{4.4} \times \dfrac{10^{7}}{10^{9}} \\
&=\dfrac{1.2}{4.4} \times 10^{7-9} \\
&=0.272 \times 10^{-2}
\end{aligned}\)

We need to move the decimal point one place to the right to put this number in scientific notation.
Thus, we must also change the exponent of \(10\).

\(
\begin{array}{l}
0.272 \times 10^{-2} \\
2.72 \times 10^{-1} \times 10^{-2} \\
2.72 \times\left(10^{-1} \times 10^{-2}\right) \\
2.72 \times\left(10^{-1+(-2)}\right) \\
2.72 \times\left(10^{-1-2}\right) \\
2.72 \times 10^{-3}
\end{array}
\)

Therefore,

\(\dfrac{1.2 \times 10^{7}}{ 4.4\times 10^{9}}=2.72 \times 10^{-3}\)