Scientific Notation

Extremely Large and Small Numbers

Large and small numbers are prevalent in our world:

In 2011 the Use National Debt was:

$15,143,572,733,215

The mass of a dust particle is approximately:

0.000000000753 kg

Scientists have created an easier way to express these extreme numbers.

Standard Form or Decimal Form

Introduction to Scientific Notation: Powers of Ten

Decimal form Powers of 10 Name1,000,000,000 109 One billion

1,000,000 106 One million1,000 103 One thousand

10 101 Ten1 100 One

0.1 10-1 One tenth0.001 10-3 One thousandth

0.000001 10-6 One millionth

Scientific Notation is based on Powers of Ten.

Numbers in Scientific Notation are written as:

c x 10n

where c is a decimal number greater than or equal to 1 and less than 10 (Typically c is rounded to 2 or 3 decimal places) and n is an integer. Below are two examples:

5 10.2 12 2013.2 10 7.035 10 3.9 10 0.45 10a b c d

Definition of Scientific Notation

5

4

1.2 10

3.5 10

Example: Which expression is written in correct Scientific

Notation? 3.9 is the only number greater than or equal to 1 and less than 10 AND the exponent is an integer

120000

0.00035

51.2 10

By converting the power of 10 to Standard Form Notation, the true value of the Scientific Notation number can be obtained.

The positive 5 makes the number bigger, by moving the decimal 5 places to the right.

The negative 4 makes the number smaller, by moving the decimal 4 places to the left.

1.2 100000

3.5 0.0001 43.5 10

Converting Standard Form to Scientific Notation

Example: Convert the following from Standard Form to Scientific Notation . Then try to find a shortcut.

1.2 0 0 0 0.1 2 3 4 5

3.50 .0 0 04 3 2 1

Rewrite each Scientific Notation in Standard Form without a calculator.

8

9

5

6

a. 1.345 10

b. 4.103 10

c. 7.8 10

d. 3.67 10

Examples

The + 8 moves the decimal 8 places to the right (bigger).

1.3 4 5 0 0 0 0 0.1 2 3 4 5 6 7 8

134,500,000

The – 9 moves the decimal 9 places to the left (smaller). 4.1030 .0 0 0 0 0 0 0 0

4 3 2 18 7 6 59

0.000000004103

The + 6 moves the decimal 6 places to the right (bigger). 3.6 7 0 0 0 0.

1 2 3 4 5 6

3,670,000

The – 5 moves the decimal 5 places to the left (smaller). 7.80 .0 0 0 0

4 3 2 15

0.000078

10,000,000,000,000131.514 1015,143,572,733,215

To convert Standard Form to Scientific Notation, find the largest power of ten such that when it divides the number, the result is between 1 and 10.

The number is made smaller by moving the decimal 13 places (power of ten needs is positive)

Converting Scientific Notation to Standard Form

Example: Convert the following from Scientific Notation to Standard Form. Then try to find a shortcut.

15143572733215.13 12 11 10 9

1.5141013 Round to 3

decimal places

8 7 6 5 4 3 2 1

.

0.0000000001107.53 100.000000000753

The number is made larger by moving the decimal 10 places (power of ten needs is negative)

0.0000000007531 2 3 4 5

7.5310-10

6 7 8 9 10

.

Rewrite each Standard Form in Scientific Notation without a calculator.

a. 245000

b. 1.78

c. 0.000722

d. 17,439,502

Examples

Move the decimal + 5 places to the left (positive since the number is made smaller).

0 .0 0 0 7 2 2.1 2 3 4

52.45 10

Already in Scientific Notation, it is a number greater than or equal to 1 and less than 10

Move the decimal + 7 places to the left (positive since the number is made

smaller).

2 4 5 0 0 0..5 4 3 2 1

71.744 10

Move the decimal - 4 places to the right (negative since

the number is made bigger).

47.22 10

1.78

1 7 4 3 9 5 0 2..5 4 3 2 17 6

4 3

1 4

a. 1.4 10 7.6 10

b. 1.2 10 4.8 10

4 3 1.4 7.6 10 10 710.64 10

In order to multiply or divide numbers in Scientific Notation, use the properties of exponents.

Examples: Simplify the following expressions

Regroup

81.064 10

Arithmetic with Scientific Notation

Scientific Notation

Simplify

1 4 1.2 4.8 10 10 30.25 10 Regroup

22.5 10 Scientific

NotationSimplify

Most scientific calculator write in large and small numbers in Scientific Notation. In TI calculators the letter E stands for x10, for example

1.5 E12 = 1.5x1012

Make sure to rewrite what the calculator displays into correct scientific notation.

Examples: Use the calculator to write the following expressions in Scientific Notation.

Scientific Notation and the Calculator

a. 78,000 2,400,000,000E1.872 14 141.872 10

b. 0.000000748 500Calculator:

E1.496 9 91.496 10 Calculator: