Notes on Percentage Changes and Scientific Notation

Percentages of percentages: absolute vs relative change

Key idea: when data are given as percentages, you can measure change in two main ways: absolute change (in percentage points) and relative change (percentage change).
Starting value and new value
- For the landline example: starting value = 90% (in 02/2004), new value = 39% (in 2020).
Absolute change (in percentage points)
- Definition:
- \text{Absolute change} = \text{new value} - \text{starting value}
- Calculation: 39\% - 90\% = -51\text{ percentage points}
- Interpretation: a decrease of 51 percentage points (the negative sign can be dropped when you explicitly state "decrease by"; if you say "change", you may say "change by -51 percentage points").
- Important nuance: when reporting absolute change you use "percentage points" rather than percent to avoid implying a relative percentage change.
- Sentence construction examples from the transcript:
- Absolute change: "The percentage points decreased by 51 percentage points." (or, equivalently, "The absolute change is 51 percentage points.")
- If you say the event decreased, you typically do not write the negative sign in the sentence.
Relative change (percent change)
- Definition:
- \text{Relative change} = \left(\frac{\text{new} - \text{start}}{\text{start}}\right) \times 100\%
- Calculation: \frac{39\% - 90\%}{90\%} \times 100\% = \frac{-51\%}{90\%} \times 100\% = -57\% (\text{approximately})
- Interpretation: a decrease of about 57% relative to the starting value.
- When expressing a relative change, you typically use the percent sign (e.g., "-57%" or "a 57% decrease").
Distinction between absolute change and relative change (informal rule mentioned in the transcript)
- If the sentence includes a percent sign, it is typically describing a relative change.
- If the sentence describes an absolute change (without a percent in the description, or explicitly mentions a change in points), you report in percentage points.
- The transcript notes this distinction to avoid confusion when a percentage is involved in an absolute change.
Homework cue mentioned in the transcript
- A question asking students to write a sentence using "percentage points" instead of "%" when describing an absolute change.
Quick practice recap
- Given: start = 90%, new = 39%
- Absolute change: \Delta_{abs} = 39\% - 90\% = -51\text{ percentage points}
- Relative change: \Delta_{rel} = \left(\frac{39\% - 90\%}{90\%}\right) \times 100\% \approx -57\%.

Scientific notation (intro and basics)

Purpose: Scientific notation is used for numbers that are extremely large or extremely small.
Core idea: A number is written as a product of a coefficient between 1 and 10 and a power of 10.
- General form:
- N = a \times 10^{\,n}
- Here:
- a is the mantissa (also called significand in newer terminology).
- n is the exponent.
Powers of 10 (recap):
- 10^n = \underbrace{10 \times 10 \times \cdots \times 10}_{n\text{ times}}
- Examples: 10^2 = 100, \; 10^3 = 1000, \; 10^0 = 1, \; 10^{-n} = \frac{1}{10^{n}}.
Scientific notation has two key traits:
- The coefficient (mantissa) is between 1 and 10: 1 \le a < 10.
- The number is expressed as a multiple of a power of 10: N = a \times 10^{n}.
Notational variants:
- A number like a \times 10^{n} can be written in computer/ calculator shorthand as a\e n or a\,e\,+\;n for positive exponents and a\,e\,-\;n for negative exponents.
- Example: 3.042 \times 10^3 can be written as 3.042e3 or 3.042e+3.
Converting a regular number to scientific notation: general steps
- Move the decimal point to place it after the first nonzero digit from the left.
- Count how many places you moved the decimal point; that count becomes the exponent (positive if you moved left, negative if you moved right).
- If the resulting coefficient is not between 1 and 10, adjust by moving the decimal and changing the exponent accordingly.
Worked examples from the lecture:
- Example A: Convert 3,042 to scientific notation.
- Move the decimal to after the first nonzero digit: 3.042, moved 3 places to the left.
- Exponent is +3 because the decimal moved left 3 places.
- Scientific notation: 3.042 \times 10^{3}.
- Example B: Convert 0.00012 to scientific notation.
- First nonzero digit is 1 (in the 5th place after the decimal).
- Move decimal so that we have 1.2, moved 4 places to the right.
- Exponent is -4 because the decimal moved to the right.
- Scientific notation: 1.2 \times 10^{-4}.
- Example C: 226 \times 10^{2} — is this already scientific notation?
- No, because 226 is not between 1 and 10.
- Normalize: move decimal to get 2.26 and adjust exponent:
  - 226 × 10^2 = 2.26 × 10^4.
- Reason: moving the decimal left by 1 place increases the coefficient division, and you add 2 to the existing exponent (2 + 2 = 4).
- Alternate notation (calculator-friendly): 3.042 × 10^3 = 3.042e3 or 3.042e+3.
Practical tip: calculator entry
- Many calculators use the EE or E button to enter scientific notation, e.g., to enter 1.2 × 10^4 you might type 1.2, press EE (or EE button), then 4 (or +4/−4 depending on the model).
Real-world example from the lecture:
- Government spending in 02/2020 reported as a huge number; converted to scientific notation:
- Value: 6.5 × 10^12 dollars, i.e., $6.5$ trillion.
- Rationale: easier to read and interpret than a long string of zeros.
Mantissa vs significand terminology
- Mantissa: traditional term for the coefficient in scientific notation.
- Significand: modern/alternative term for the coefficient.
- The instructor prefers mantissa, but notes that significand is the newer standard term in many textbooks.
- Note the two terms are synonyms in this context.
Multiplication and division rules in scientific notation
- Multiplication:
- If N1 = a \times 10^{m} and N2 = b \times 10^{n}, then
  - N1 \times N2 = (a\cdot b) \times 10^{m+n}
- After multiplying mantissas, you may need to normalize so the coefficient is between 1 and 10, adjusting the exponent accordingly.
- Example from the transcript:
  - (4 \times 10^{5}) \times (6 \times 10^{2}) = 24 \times 10^{7}
  - Normalize: move decimal to get 2.4 and increase exponent by 1: 2.4 \times 10^{8}.
- Division (opposite of multiplication):
- If N1 = a \times 10^{m} and N2 = b \times 10^{n}, then
  - \frac{N1}{N2} = \left(\frac{a}{b}\right) \times 10^{m-n}
- Normalize if needed (to keep coefficient between 1 and 10).
- Example from the transcript:
  - (4 \times 10^{5}) / (6 \times 10^{2}) = \left(\frac{4}{6}\right) \times 10^{3} \approx 0.666\bar{6} \times 10^{3}
  - Normalize: move decimal so coefficient is between 1 and 10: 6.67 \times 10^{2}. (approximate)
Orders of magnitude
- Definition (rough): the power of 10 that describes the scale of a number, i.e., the exponent in scientific notation.
- Same order of magnitude when exponents are equal; different orders of magnitude when exponents differ.
- Examples from the lecture:
- 2.0 × 10^9 (2,000,000,000) and 8.5 × 10^9 (8,500,000,000) have the same order of magnitude (both in the billions), even though one is larger by a factor ~4.25.
- 5.6 × 10^4 (56,000) and 2.0 × 10^9 are in different orders of magnitude; here the exponent difference is 9 − 4 = 5, so the second number is five orders of magnitude bigger.
Quick conceptual check: orders of magnitude difference
- If you change the exponent by Δn, you change the size by a factor of 10^Δn.
- Example: A number with exponent 4 and another with exponent 9 differ by 9 − 4 = 5 orders of magnitude (a factor of 10^5).
A concrete application: estimating total trash production (per person to city scale)
- Given:
- Each person produces about 2.4 pounds of trash. This is equivalent to 0.0012 tons per person (since 1 ton = 2000 pounds).
  - 2.4\text{ lb} = 0.0012\text{ tons}
- Population: 8,800,000 people, which is 8.8 \times 10^{6} people.
- Convert population to scientific notation: 8.8 \times 10^{6}.
- Total trash in tons: multiply per-person tons by population:
- (8.8 \times 10^{6}) \times (1.2 \times 10^{-3}) = (8.8 \times 1.2) \times 10^{6-3} = 10.56 \times 10^{3} = 1.056 \times 10^{4} \text{ tons}
- Interpretation: about 10,560 tons of trash per day for the city (to the nearest thousand, the order is thousands).
- Comparison with a friend’s estimate (225 tons):
- 225 tons ≈ 2.25 × 10^2 tons, which is far smaller than the order of 10^4 tons.
- Conclusion from the lecture: your friend’s estimate is not reasonable; the actual order of magnitude is on the order of 10^3 to 10^4 tons per day, i.e., thousands to ten-thousands of tons per day.
- Language cue used in the lecture:
- “On the order of 10^3” to describe the approximate scale of the total trash.

Quick reference: key formulas

Absolute change (in percentage points):
- \Delta_{abs} = \text{new} - \text{start}
Relative change (percent change):
- \Delta_{rel} = \left(\frac{\text{new} - \text{start}}{\text{start}}\right) \times 100\%
Percentage-point language cue:
- If reporting a change in percentage points, use the term "percentage points" (no negative sign when saying "decreased by" or similar).
Scientific notation basics:
- N = a \times 10^{n}, \quad 1 \le a < 10, \; n \in \mathbb{Z}
- Entering notation in calculators: a\e n\;\text{or}\;a\text{e}+n,\;a\text{e}-n
Multiplication in scientific notation:
- (a\times 10^{m}) \cdot (b\times 10^{n}) = (a b) \times 10^{m+n}
- Normalize to keep mantissa in [1, 10], adjust exponent accordingly.
Division in scientific notation:
- \frac{a\times 10^{m}}{b\times 10^{n}} = \left(\frac{a}{b}\right) \times 10^{m-n}
- Normalize if needed.
Orders of magnitude intuition:
- Exponent difference Δn corresponds to a factor of 10^{\Delta n} in size.

Quick glossary

Absolute change: the numeric change in a quantity, measured in the same units or in percentage points when dealing with percentages.
Relative change: change expressed as a percentage of the original value.
Percentage points: unit for absolute changes in percentages; helps distinguish from percent changes.
Mantissa (significand): the coefficient in scientific notation between 1 and 10.
Exponent: the power of 10 in scientific notation.
Scientific notation: a way to express numbers as a × 10^n with a in [1, 10).
Ordinal magnitude (order of magnitude): the scale of a number described by its exponent in scientific notation.
E notation: calculator shorthand for scientific notation (e stands for exponent).