Scientific Notation

How do we use mathematical shorthand to write very large and very small numbers? We focus on the most important digits, and use an abbreviation to avoid writing out the insignificant digits.

Usually the only significant information is in the first couple of digits and the location in place value that they occupy.

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
  • Note that you already use scientific notation when you speak about large numbers
    • You don’t say one zero zero zero zero zero zero
    • You say one million

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

Scientific Notation English SI prefix Symbol Mnemonic
10^{24} Septillion yotta Y yodeling
10^{21} Sextillion zetta Z zebras
10^{18} Quintillion exa E expect
10^{15} Quadrillion peta P patient
10^{12} Trillion tera T teachers
10^{9} Billion giga G giving
10^{6} Million mega M metric
10^{3} Thousand kilo k knowledge