Subsection 1.1.1 Prep Text: Scientific Notation
To wield scientific notation properly, follow these arcane guidelines:
Find the coefficient. Move the decimal point so the number becomes at least 1 but less than 10. This becomes the coefficient. For example, 32,700 becomes "3.27".
Count the decimal steps. Count how many places you moved the decimal. That number becomes the exponent on your power of ten. If you moved the decimal to the left, the exponent is positive. If you moved it to the right, the exponent is negative.
Check your form. The final expression should look like \(a \times 10^n\text{,}\) where \(1 \leq |a| < 10\) and \(n\) is an integer.
Checkpoint 1.1.1.
Convert 32,700 to scientific notation.
Hint.
The decimal point would need to move to the left by 4 places.
Answer.
Solution.
To write 32,700 in scientific notation, move the decimal point four places to the left to place it after the first nonzero digit. This gives "3.27". Since the decimal moved left by four places, the exponent on 10 is 4.
So, \(32,700 = 3.27 \times 10^4\text{.}\)
Checkpoint 1.1.2.
Convert 0.0000089 to scientific notation.
Hint.
The decimal point would need to move to the right by 6 places.
Answer.
Solution.
To write 0.0000089 in scientific notation, move the decimal point six places to the right to place it after the first nonzero digit. This gives "8.9". Because the decimal moved right by six places, the exponent on 10 is -6.
So, \(0.0000089 = 8.9 \times 10^{-6}\text{.}\)
Why Use It?
Imagine trying to write the mass of the Earth (~\(5.97 \times 10^{24}\) kg) or the mass of an electron (~\(9.1 \times 10^{-31}\) kg) without magical shorthand! Scientific notation saves parchment, keeps calculations neat, and makes comparison of large and small values easier.
Bonus Trick: Comparing Powers
Scientific notation reveals orders of magnitude. Two numbers differ by one order of magnitude for every tenfold difference. A dragon’s hoard of \(4.2 \times 10^6\) gold coins is about three orders of magnitude larger than a goblin’s pouch of \(1.7 \times 10^3\text{.}\) 3 was obtained by subtracting the exponents (6 - 3 = 3).
Likewise, a nanobot’s processor that uses \(3.0 \times 10^{-9}\) joules per operation is about two orders of magnitude smaller than a clunky space probe core drawing \(1 \times 10^{-7}\) joules per operation. 2 was obtained by subtracting the exponents (-9 - (-7) = -2).
A Parting Word
With scientific notation in your spellbook, you can navigate the vast realms of physics, chemistry, and cosmic measurement. Write your numbers with care, check your powers of ten, and let no zero go unwielded.
