Scientific Notation

Scientific Notation

• Allows us to compactly write very large or very small numbers

• Creates a shorthand that allows us to simplify calculation and avoid errors

Orders of magnitude

• Powers of ten

• Place value representation

• Operating on numbers with scientific notation

Representing large numbers

There are different ways to express scientific notation

• On paper $1.23 \times 10^4$

• On a calculator 1.23 (EE key) 4

• On a computer 1.23E4

A very large number

• Avogadro's Number

• $6.02 \times 10^{23}$
• $10^{1}$ = 10
• $10^{2} = 10 \times 10$
• $10^{3} = 10 \times 10 \times 10$

A very small number

• Gravitational Constant

• $6.67 \times 10^{-11} (m^3 kg^{-1} s^{-2})$
• $10^{-1} = \frac{1}{10}$
• $10^{-2} = \frac{1}{10 \times 10}$
• $10^{-3} = \frac{1}{10 \times 10 \times 10}$

Operations

• Addition

• Subtraction

• Multiplication

• Division

Multiplying and dividing large numbers

$10^a \cdot 10^b = 10^{a+b}$

$c \cdot 10^a \cdot d \cdot 10^b = c \cdot d \cdot 10^{a+b}$

$\frac{10^a}{10^b} = 10^{a-b}$

$\frac{c \cdot 10^a}{d \cdot 10^b} = \frac{c}{d} \cdot 10^{a-b}$

You can combine these rules if you are multiplying and dividing several numbers in scientific notation.

Common Powers

Notice that these prefixes correspond to english language quantities you are already familiar with. You often express large numbers verbally, so there isn't much new to learn to use metric prefixes or scientific notation.

For example, 1 billion joules is the same as a gigajoule is the same as $1 \cdot 10^{9}$ joules is the same as 1,000,000,000 joules.

A more complete list can be found here:

Wikipedia Metric Prefixes

kilo	thousand	10$^3$
mega	million	10$^6$
giga	billion	10$^9$
tera	trillion	10$^{12}$
peta	quadrillion	10$^{15}$
exa	quintillion	10$^{18}$