IB · Unit 1 · 1B Scientific Notation
One digit in front of the decimal, times a power of 10. If the coefficient is not yet between 1 and 10, fix it.
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\(x^2 \cdot x^3 = x^5\)
\((x^2)^3 = x^6\)
3−2 = 19
Standard form
Scientific notation and IB standard form are the same thing: a × 10ᵏ with 1 ≤ a < 10 and k ⊂ ℤ.
Big numbers
Move the decimal left until one digit is in front. k is the number of places you moved.
45 000 = \(4.5 \times 10^{4}\)
Small numbers
Move the decimal right until one digit is in front. k is negative.
0.00047 = \(4.7 \times 10^{-4}\)
Already almost standard
If the coefficient is 10 or more, or less than 1, adjust it.
\(12.08 \times 10^{5}\)
\(1.208 \times 10^{1} \times 10^{5}\)
\(1.208 \times 10^{6}\)
When a question says write the answer in the form a × 10ᵏ where 1 ≤ a < 10 and k ⊂ ℤ, that is the IB way of saying write the answer in scientific notation.
k ⊂ ℤ means k is an integer (k is in the set of integers).
Answers will look like 3.2E4. That means \(3.2 \times 10^4\). Write the × 10 form on paper.
One move of the decimal. Coefficient already between 1 and 10, or a whole-number power of 10.
Move the decimal 4 places left so one digit is in front.
45 000 = \(4.5 \times 10^{4}\)
Answer: \(4.5 \times 10^{4}\)
Move the decimal 3 places right.
3700
Answer: 3700
Move the decimal 5 places right.
810 000
Answer: 810 000
Move the decimal 3 places left.
0.0024
Answer: 0.0024
Move the decimal 3 places left.
5600 = \(5.6 \times 10^{3}\)
Answer: \(5.6 \times 10^{3}\)
Move the decimal 1 place left.
91 = \(9.1 \times 10^{1}\)
Answer: \(9.1 \times 10^{1}\)
More digits, or a decimal that is already less than 1.
Move the decimal 4 places right so one digit is in front.
0.00047 = \(4.7 \times 10^{-4}\)
Answer: \(4.7 \times 10^{-4}\)
Move the decimal 5 places right.
\(8.3 \times 10^{-5}\)
Answer: \(8.3 \times 10^{-5}\)
Move the decimal 2 places left.
\(4.126 \times 10^{2}\)
Answer: \(4.126 \times 10^{2}\)
Move the decimal 1 place left.
0.903
Answer: 0.903
Move the decimal 2 places right.
\(7.05 \times 10^{-2}\)
Answer: \(7.05 \times 10^{-2}\)
Move the decimal 5 places left.
\(5.803 \times 10^{5}\)
Answer: \(5.803 \times 10^{5}\)
The coefficient is not yet between 1 and 10. Adjust it, and change the power of 10 to match.
12.08 is bigger than 10, so move the decimal 1 place left.
\(12.08 \times 10^{5}\)
\(1.208 \times 10^{1} \times 10^{5}\)
\(1.208 \times 10^{6}\)
Answer: \(1.208 \times 10^{6}\)
0.46 is less than 1, so move the decimal 1 place right.
\(0.46 \times 10^{-3}\)
\(4.6 \times 10^{-1} \times 10^{-3}\)
\(4.6 \times 10^{-4}\)
Answer: \(4.6 \times 10^{-4}\)
Move the decimal 8 places right.
0.0000000901
\(9.01 \times 10^{-8}\)
Answer: \(9.01 \times 10^{-8}\)
64.2 is bigger than 10, so move the decimal 1 place left.
\(64.2 \times 10^{-5}\)
\(6.42 \times 10^{1} \times 10^{-5}\)
\(6.42 \times 10^{-4}\)
Answer: \(6.42 \times 10^{-4}\)
0.0039 is less than 1, so move the decimal 3 places right.
\(0.0039 \times 10^{8}\)
\(3.9 \times 10^{-3} \times 10^{8}\)
\(3.9 \times 10^{5}\)
Answer: \(3.9 \times 10^{5}\)
Move the decimal 7 places right.
0.000000305
\(3.05 \times 10^{-7}\)
Answer: \(3.05 \times 10^{-7}\)
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