Scientific Notation

Scientific notation is a compact way to write very large or very small numbers using powers of 10. Instead of writing out 93,000,000 miles (the distance from Earth to the Sun), scientists write $9.3 \times 10^7$. Instead of writing 0.000000001 meters (the size of a molecule), they write $1 \times 10^{-9}$. Scientific notation keeps numbers manageable and makes comparisons easy.

The Format

A number in scientific notation has two parts:

$a \times 10^n$

where:

• $a$ is called the coefficient — a number between 1 and 10 (including 1, but not including 10): $1 \leq a < 10$
• $10^n$ is a power of 10 — the exponent $n$ is a positive or negative integer

Examples of valid scientific notation: $3.5 \times 10^4$, $1.0 \times 10^{-6}$, $8.21 \times 10^{12}$

Not valid scientific notation: $35 \times 10^3$ (coefficient is 35, which is not between 1 and 10) or $0.5 \times 10^8$ (coefficient is less than 1)

Converting Standard Form to Scientific Notation

For Large Numbers (positive exponent)

Step 1: Move the decimal point to the left until you have a number between 1 and 10.

Step 2: Count how many places you moved the decimal. That count becomes the positive exponent.

Example 1: Convert 4,500,000 to Scientific Notation

Step 1: Move the decimal from the end of 4,500,000. to between the 4 and 5: $4.500000$.

Step 2: The decimal moved 6 places to the left.

$4{,}500{,}000 = 4.5 \times 10^6$

Answer: $4.5 \times 10^6$

Example 2: Convert 328,000 to Scientific Notation

Move the decimal 5 places left: $3.28$.

$328{,}000 = 3.28 \times 10^5$

Answer: $3.28 \times 10^5$

For Small Numbers (negative exponent)

Step 1: Move the decimal point to the right until you have a number between 1 and 10.

Step 2: Count how many places you moved the decimal. That count becomes the negative exponent.

Example 3: Convert 0.00072 to Scientific Notation

Step 1: Move the decimal right until you reach $7.2$ — that is 4 places.

$0.00072 = 7.2 \times 10^{-4}$

Answer: $7.2 \times 10^{-4}$

Example 4: Convert 0.0000056 to Scientific Notation

Move the decimal 6 places right to get $5.6$.

$0.0000056 = 5.6 \times 10^{-6}$

Answer: $5.6 \times 10^{-6}$

Converting Scientific Notation to Standard Form

For Positive Exponents (large numbers)

Move the decimal to the right by the number of places indicated by the exponent. Fill with zeros as needed.

Example 5: Convert $6.02 \times 10^{5}$ to Standard Form

Move the decimal 5 places right:

$6.02000 \rightarrow 602{,}000$

Answer: $602{,}000$

Example 6: Convert $1.5 \times 10^{8}$ to Standard Form

Move the decimal 8 places right:

$1.50000000 \rightarrow 150{,}000{,}000$

Answer: $150{,}000{,}000$

For Negative Exponents (small numbers)

Move the decimal to the left by the number of places indicated by the exponent. Fill with zeros.

Example 7: Convert $3.7 \times 10^{-3}$ to Standard Form

Move the decimal 3 places left:

$3.7 \rightarrow 0.0037$

Answer: $0.0037$

Example 8: Convert $9.1 \times 10^{-7}$ to Standard Form

Move the decimal 7 places left:

$9.1 \rightarrow 0.00000091$

Answer: $0.00000091$

Comparing Numbers in Scientific Notation

To compare numbers in scientific notation:

Step 1: Compare the exponents. The number with the larger exponent is larger (for positive values).

Step 2: If the exponents are equal, compare the coefficients.

Example 9: Which Is Larger: $3.2 \times 10^6$ or $8.7 \times 10^4$?

The exponents are 6 and 4. Since $6 > 4$:

$3.2 \times 10^6 > 8.7 \times 10^4$

Even though 8.7 is a bigger coefficient than 3.2, the power of 10 dominates. $3.2 \times 10^6 = 3{,}200{,}000$ versus $8.7 \times 10^4 = 87{,}000$.

Answer: $3.2 \times 10^6$ is larger.

Example 10: Which Is Larger: $4.5 \times 10^{-3}$ or $4.5 \times 10^{-5}$?

Both coefficients are 4.5. Compare the exponents: $-3 > -5$.

$4.5 \times 10^{-3} = 0.0045 \qquad 4.5 \times 10^{-5} = 0.000045$

Answer: $4.5 \times 10^{-3}$ is larger (closer to zero means smaller for negatives, but these are positive numbers — a less-negative exponent gives a larger value).

Example 11: Ordering from Smallest to Largest

Order: $2.1 \times 10^{-2}$, $5.0 \times 10^{3}$, $9.9 \times 10^{-5}$, $1.3 \times 10^{3}$

Step 1: Group by exponent.

• Exponent $-5$: $9.9 \times 10^{-5} = 0.000099$ (smallest)
• Exponent $-2$: $2.1 \times 10^{-2} = 0.021$
• Exponent $3$: $1.3 \times 10^{3} = 1{,}300$ and $5.0 \times 10^{3} = 5{,}000$

Step 2: Within exponent 3, compare coefficients: $1.3 < 5.0$.

Answer (smallest to largest): $9.9 \times 10^{-5}$, $2.1 \times 10^{-2}$, $1.3 \times 10^{3}$, $5.0 \times 10^{3}$

Real-World Application: Nursing — Drug Dosages

Medication concentrations often involve very small numbers. A drug label might state:

Epinephrine concentration: 0.001 g/mL (which is $1 \times 10^{-3}$ g/mL, or equivalently 1 mg/mL)

If a doctor orders 0.5 mg, the nurse calculates:

$0.5 \text{ mg} = 5 \times 10^{-4} \text{ g}$

Using scientific notation prevents confusion between milligrams, micrograms, and grams — a critical safety concern in healthcare.

Common medical prefixes and their powers of 10:

Real-World Application: Electrician — Component Values

Electricians and electronics technicians use scientific notation for capacitor and resistor values:

• A 470 nF capacitor: $470 \times 10^{-9} = 4.7 \times 10^{-7}$ farads
• A 2.2 M-ohm resistor: $2{,}200{,}000 = 2.2 \times 10^{6}$ ohms
• A 47 $\mu$F capacitor: $0.000047 = 4.7 \times 10^{-5}$ farads

Reading component values from a circuit diagram requires fluency with scientific notation.

Common Mistakes to Avoid

1. Coefficient outside the range 1 to 10. $25 \times 10^3$ is not proper scientific notation. It should be $2.5 \times 10^4$.

2. Moving the decimal the wrong direction. For large numbers, move left (positive exponent). For small numbers, move right (negative exponent).

3. Comparing only the coefficients. $9.9 \times 10^2 = 990$ is much smaller than $1.1 \times 10^5 = 110{,}000$. The exponent matters more than the coefficient.

4. Forgetting that $10^0 = 1$. The number $5.0$ in scientific notation is $5.0 \times 10^0$.

5. Miscounting decimal places. For $0.00072$, the decimal moves 4 places (not 3) to reach $7.2$. Count carefully.

Practice Problems

Key Takeaways

• Scientific notation writes numbers as $a \times 10^n$ where $1 \leq a < 10$.
• Positive exponents represent large numbers; negative exponents represent small numbers.
• To convert to scientific notation, move the decimal until $a$ is between 1 and 10, then count the places moved.
• To convert from scientific notation, move the decimal right (positive $n$) or left (negative $n$).
• When comparing, the exponent determines size first; use the coefficient only when exponents are equal.
• Scientific notation is essential in nursing (drug concentrations) and electrical work (component values).

Return to Pre-Algebra for more topics in this section.