Section 1.1 Prep
This preparatory section will equip you with the knowledge and skills to wield scientific notation with confidence and precision. Through a combination of instructional video, explanatory text, and practice questions, you will become adept at all things scientific notation.
Subsection 1.1.1 Prep Videos: Scientific Notation
Watch the following video(s) to get an introduction to scientific notation.
Subsection 1.1.2 Prep Text: Scientific Notation
Read the following text to learn about scientific notation. Don’t forget to try the checkpoint questions along the way!
Subsubsection 1.1.2.1 What Is Scientific Notation?
Definition 1.1.1.
Scientific Notation A standardized way to write very large or very small numbers using a coefficient between \(1\) and \(10\) multiplied by a power of ten. In scientific notation, a number has the form \(a \times 10^{n}\text{,}\) where \(1 \le |a| < 10\) and \(n\) is an integer. The exponent \(n\) indicates how many places the decimal point has been moved.
Subsubsection 1.1.2.2 Writing Numbers in 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 \(10\text{.}\) This becomes the coefficient. For example, \(32700\) becomes \(3.27\text{.}\)
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 1.1.2.
Convert \(32700\) to scientific notation.
Hint.
The decimal point would need to move to the left by \(4\) places.
Answer.
Solution.
To write \(32700\) in scientific notation, move the decimal point four places to the left to place it after the first nonzero digit. This gives \(3.27\text{.}\) Since the decimal moved left by four places, the exponent on \(10\) is \(4\text{.}\)
So, \(32700 = 3.27 \times 10^4\text{.}\)
Checkpoint 1.1.3.
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 \(8.9\text{.}\) Because the decimal moved right by six places, the exponent on the \(10\) is \(-6\text{.}\)
So, \(0.0000089 = 8.9 \times 10^{-6}\text{.}\)
Subsubsection 1.1.2.3 Orders of Magnitude
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 4 orders of magnitude larger than a goblin’s pouch of \(1.7 \times 10^2\) coins. The order of magnitude, \(4\text{,}\) was obtained by subtracting the exponents: \(6 - 2 = 4\)
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. The two for order of magnitude was obtained by subtracting the original exponents: \(-9 - (-7) = -2\)
Checkpoint 1.1.4.
How many orders of magnitude larger is \(9.90 \times 10^{-7}\) compared to \(3.76 \times 10^{-9}\text{.}\)
Hint.
Subtract the exponents of the base \(10\text{.}\)
Answer.
Solution.
Start with the exponent of the number we want to know how much larger: \(-7\text{.}\) Then subtract the exponent of the number we are comparing too: \(-9\text{.}\)
\(-7 - (-9) = 2\)
So, \(9.90 \times 10^{-7}\) is \(2\) orders of magnitude larger than \(3.76 \times 10^{-9}\text{.}\)
Checkpoint 1.1.5.
How many orders of magnitude smaller is \(3.21 \times 10^{1}\) compared to \(7.94 \times 10^{8}\text{.}\)
Hint.
Subtract the exponents of the base \(10\text{.}\)
Answer.
Solution.
Start with the exponent of the number we want to know how much smaller: \(1\text{.}\) Then subtract the exponent of the number we are comparing too: \(8\text{.}\)
\(1 - 8 = - 7\)
So, \(3.21 \times 10^{1}\) is \(7\) orders of magnitude smaller than \(7.94 \times 10^{8}\text{.}\)
Subsubsection 1.1.2.4 Why Use Scientific Notation?
Imagine trying to write the mass of the Earth (about \(5.97 \times 10^{24}\) kilograms), or the mass of an electron (about \(9.11 \times 10^{-31}\) kilograms) without magical shorthand! Scientific notation saves parchment, keeps calculations neat, and makes comparison of large and small values easier.
Subsubsection 1.1.2.5 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.
Subsection 1.1.3 Prep Questions: Scientific Notation
Convert standard notation to scientific notation.
Checkpoint 1.1.6.
Convert \(0.0072\) to scientific notation.
\(7.2 \times 10^{-3}\)
Correct! \(0.0072 = 7.2 \times 10^{-3}\)
\(72 \times 10^{-4}\)
Your targeting system glitched, try again. While \(72 \times 10^{-4} = 0.0072\text{,}\) the coefficient must be greater than or equal to \(1\) and less than \(10\) for scientific notation.
\(7.2 \times 10^{3}\)
Your enchanted click missed the mark, try again. The decimal moved left, so the power should be negative.
\(7.2 \times 10^{-2}\)
Your digital blade missed the mark, try again. Try counting the decimal places again.
Checkpoint 1.1.7.
Convert \(0.00000702\) to scientific notation.
\(7.02 \times 10^{-6}\)
Correct! \(0.00000702 = 7.02 \times 10^{-6}\)
\(7.0 \times 10^{-6}\)
Your targeting system glitched, try again. You left out a digit—don’t forget the \(2\text{.}\)
\(7.02 \times 10^{6}\)
Your enchanted click missed the mark, try again. The decimal moved left, so the exponent should be negative.
\(7.02 \times 10^{-5}\)
Your digital blade missed the mark, try again. Try counting the decimal places again.
Checkpoint 1.1.8.
Convert \(0.70\) to scientific notation.
\(7.0 \times 10^{-1}\)
Correct! \(0.70 = 7.0 \times 10^{-1}\)
\(0.70 \times 10^{0}\)
Your targeting system glitched, try again. While \(0.70 \times 10^{0} = 0.70\text{,}\) the coefficient must be greater than or equal to \(1\) and less than \(10\) for scientific notation.
\(7.0 \times 10^{1}\)
Your enchanted click missed the mark, try again. The decimal moved left, so the exponent should be negative.
\(0.70 \times 10^{1}\)
Your digital blade missed the mark, try again. That equals \(7.0\text{,}\) not \(0.70\text{.}\)
Checkpoint 1.1.9.
Convert \(4242\) to scientific notation.
\(4.242 \times 10^{3}\)
Correct! \(4,242 = 4.242 \times 10^{3}\)
\(4.242 \times 10^{4}\)
Your targeting system glitched, try again. Try counting the decimal places again.
\(4.242 \times 10^{-3}\)
Your enchanted click missed the mark, try again. The decimal moved right, so the exponent should be positive.
\(4.242 \times 10^{-4}\)
Your digital blade missed the mark, try again. Count your decimal places carefully, and remember if the decimal moved right then exponent should be positive.
Checkpoint 1.1.10.
Convert \(10.0\) to scientific notation.
\(1.0 \times 10^{1}\)
Correct! \(10.0 = 1.0 \times 10^{1}\)
\(1.0 \times 10^{0}\)
Your targeting system glitched, try again. That equals \(1.0\text{,}\) not \(10.0\text{.}\)
\(10.0 \times 10^{0}\)
Your enchanted click missed the mark, try again. While \(10.0 \times 10^{0} = 10.0\text{,}\) the coefficient must be greater than or equal to \(1\) and less than \(10\) for scientific notation.
\(1.0 \times 10^{-1}\)
Your digital blade missed the mark, try again. The decimal moved right, so the exponent should be positive.
Checkpoint 1.1.11.
Convert \(1.0\) to scientific notation.
\(1.0 \times 10^{0}\)
Correct! \(1.0 = 1.0 \times 10^{0}\)
\(10. \times 10^{-1}\)
Your targeting system glitched, try again. While \(10.
\times 10^{0} = 10.0\text{,}\) the coefficient must be greater than or equal to \(1\) and less than \(10\) for scientific notation.
\(0.10 \times 10^{1}\)
Your enchanted click missed the mark, try again. While \(0.10 \times 10^{0} = 1.0\text{,}\) the coefficient must be greater than or equal to \(1\) and less than \(10\) for scientific notation.
\(1.0 \times 10^{1}\)
Your digital blade missed the mark, try again. That equals \(10.0\text{,}\) not \(1.0\text{.}\)
Convert scientific notation to standard notation.
Checkpoint 1.1.12.
Convert \(6.23 \times 10^{-2}\) to standard notation.
\(0.0623\)
Correct! \(6.23 \times 10^{-2} = 0.0632\)
\(0.00623\)
Your targeting system glitched, try again. Try counting the decimal places again.
\(6230\)
Your enchanted click missed the mark, try again. Count your decimal places carefully and recall that if the exponent is negative the decimal should move to the left.
\(623\)
Your digital blade missed the mark, try again. The exponent is negative, so the decimal should move to the left.
Checkpoint 1.1.13.
Convert \(42 \times 10^{3}\) to standard notation.
\(42000\)
Correct! \(42 \times 10^{3} = 42000\)
\(4200\)
Your targeting system glitched, try again. Try counting the decimal places again.
\(0.00042\)
Your enchanted click missed the mark, try again. Count your decimal places carefully and recall that if the exponent is positive the decimal should move to the right.
\(0.042\)
Your digital blade missed the mark, try again. The exponent is positive, so the decimal should move to the right.
Checkpoint 1.1.14.
Convert \(5.02 \times 10^{1}\) to standard notation.
\(50.2\)
Correct! \(5.02 \times 10^{1} = 50.2\)
\(502\)
Your targeting system glitched, try again. Try counting the decimal places again.
\(0.0502\)
Your enchanted click missed the mark, try again. Count your decimal places carefully and recall that if the exponent is positive the decimal should move to the right.
\(0.502\)
Your digital blade missed the mark, try again. The exponent is positive, so the decimal should move to the right.
Checkpoint 1.1.15.
Convert \(5122 \times 10^{-3}\) to standard notation.
\(5.122\)
Correct! \(5122 \times 10^{-3} = 5.122\)
\(51.22\)
Your targeting system glitched, try again. Try counting the decimal places again.
\(512200\)
Your enchanted click missed the mark, try again. Count your decimal places carefully and recall that if the exponent is negative the decimal should move to the left.
\(5122000\)
Your digital blade missed the mark, try again. The exponent is negative, so the decimal should move to the left.
Checkpoint 1.1.16.
Convert \(3 \times 10^{1}\) to standard notation.
\(30\)
Correct! \(3 \times 10^{1} = 30\)
\(300\)
Your targeting system glitched, try again. Try counting the decimal places again.
\(0.003\)
Your enchanted click missed the mark, try again. Count your decimal places carefully and recall that if the exponent is positive the decimal should move to the right.
\(0.03\)
Your digital blade missed the mark, try again. The exponent is positive, so the decimal should move to the right.
Checkpoint 1.1.17.
Convert \(7.55 \times 10^{0}\) to standard notation.
\(7.55\)
Correct! \(7.55 \times 10^{0} = 7.55\)
\(0\)
Your targeting system glitched, try again. Recall any nonzero number raised to \(0\) is equal to \(1\text{.}\)
\(75.5\)
Your enchanted click missed the mark, try again. Recall any nonzero number raised to \(0\) is equal to \(1\text{.}\)
\(0.755\)
Your digital blade missed the mark, try again. Recall any nonzero number raised to \(0\) is equal to \(1\text{.}\)
