Precalculus

Dr. Lalitha Subramanian
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Aksharam

Chapter P - Preliminary Concepts

P9: Scientific Notation

Concept of writing any positive real number in the form of scientific notation was developed to help calculations of very large or very small quantities, frequently encountered in Physics, Chemistry, Biology, Statistics, Microbiology, etc. This notation states the number in the form $k \times 10^m$, where $1 \leq k \lt 10$ and $m$ is an integer.

Example 1:

$65,000,000=6.5 \times 10^7$

Example 2:

$0.000 000 000 96 = 9.6 \times 10^{-10}$

For very large numbers, place the decimal after the first digit from the left. Then, count the number of remaining digits in the number. This number becomes the positive exponent of $10$.

For very small numbers, place the decimal after the first non-zero digit. Then, count the number of digits preceding the decimal point that you have placed, until the original decimal point. This number becomes the negative exponent of $10$.

Example 3:

Write the number $387,000,000,000,000$ in scientific notation.

Solution: $387,000,000,000,000=3.87 \times 10^{14}$

Example 4:

Write $0.000 000 000 000 000 163$

Solution: $0.000 000 000 000 000 163=1.63 \times 10^{-16}$

Example 5:

$5.138 \times 10^{11} = 513, 800, 000, 000$

Example 6:

$2.93 \times 10^{-9} = 0.000 000 00298$

Multiplication and division problems of very large or very small number can be done with ease, using scientific notation.

Example 7:

Simplify: $(45,000,000)(0.000 000 238)$

Solution: We first write the numbers in scientific notation.Then, multiply the numbers, and add the exponents using laws of exponents. Finally, convert the result into scientific notation or write it in decimal form as required. \begin{eqnarray} (45,000,000)(0.000 000 0238) &=& (4.5 \times 10^7)(2.38 \times 10^8)\\ &=& (4.5 \times 2.38)(10^7 \times 10^8)\\ &=& 10.71 \times 10^{15}\\ &=& 1.071 \times 10^{16}\\ \end{eqnarray}

Example 8:

Simplify: $\frac{(35,000,000)(0.000 000 000 021)}{14,100,000 000}$

Solution: \begin{eqnarray} \frac{(35,000,000)(0.000 000 000 021)}{14,100,000 000} &=& \frac{(3.5 \times 10^7)(2.1 \times 10^{-11})}{1.41 \times 10^{10}}\\ &=& \frac{(3.5)(2.1)}{1.41} \times 10^{7-11-10}\\ &=& 5.21 \times 10^{-14}\\ \end{eqnarray}

If there are more than 4 non-zero digits in the number, then, approximate it to three significant digits. This means, one integral part and two decimal digits.

Precalculus

Chapter P - Preliminary Concepts

P9: Scientific Notation

Example 1:

Example 2:

Example 3:

Example 4:

Example 5:

Example 6:

Example 7:

Example 8:

Practice Problems

$5.52 \times 10^8$

$7.6 \times 10^{-5}$

$0.00000333$

$7,845,000,000$

$3.68 \times 10^7$

$0.0000000123$

Federal: $\$3,950, 000, 000, 000, 000$; State: $\$23,400,000,000$; Local: $\$339,000$

$2.3 \times 10^{-11}$

$4.47 \times 10^9$ miles

$0.000 000 000 000 000 000 001655$ mg.