00-12 — Scientific Notation and Standard Form

Phase: 0 — Arithmetic & Number Foundations Subject: 00-12 Prerequisites: 00-03 — Decimals, 00-11 — Laws of Indices (Exponents) Next subject: 00-13 — Number Sequences and Patterns

Learning Objectives

By the end of this subject, you will be able to:

1. Convert any number between standard decimal form and scientific notation (standard form): a × 10ⁿ where 1 ≤ a < 10
2. Identify the order of magnitude of a number from its scientific notation exponent
3. Perform multiplication, division, addition, and subtraction with numbers expressed in scientific notation
4. Compare and order numbers of vastly different magnitudes using scientific notation
5. Apply scientific notation to real-world problems involving very large quantities (astronomical distances) and very small quantities (atomic scales)

Core Content

1. What Is Scientific Notation?

Scientific notation (also called standard form) is a compact way to write very large or very small numbers using powers of 10:

a × 10ⁿ

Where: - a is the mantissa (or coefficient): a number such that 1 ≤ a < 10 - n is an integer (positive, negative, or zero) — the exponent or order of magnitude

Why use it? - The mass of the Earth is approximately 5,970,000,000,000,000,000,000,000 kg — that's 24 zeros. In scientific notation: 5.97 × 10²⁴ kg. Much easier to read, write, and calculate with. - The diameter of a hydrogen atom is about 0.0000000001 m. In scientific notation: 1 × 10⁻¹⁰ m.

⚠️ THIS IS CRITICAL — Scientific notation is the standard way scientists, engineers, and mathematicians express very large and very small quantities. It appears in physics, chemistry, biology, astronomy, and computing. In later mathematics, it connects directly to logarithms and exponential functions.

2. Converting Large Numbers (Positive Exponent)

For numbers ≥ 10: move the decimal point left until exactly one non-zero digit remains to its left. Count the moves — that's the exponent.

Algorithm: 1. Place the decimal point after the first non-zero digit 2. Count how many places you moved the decimal point 3. The exponent is that count (positive, because the original number was ≥ 10)

Example 1: Convert 45,600 to scientific notation.

$45,600 = 4.56 × 10⁴
         ↑
Move decimal 4 places left: 45600. → 4.5600
$

Example 2: Convert 9,030,000 to scientific notation.

$9,030,000 = 9.03 × 10⁶
Move decimal 6 places left.
$

Example 3: Convert 325 to scientific notation.

$325 = 3.25 × 10²
Move decimal 2 places left: 325. → 3.25
$

Example 4: Convert 7,000,000,000 to scientific notation.

$7,000,000,000 = 7 × 10⁹
$

Key check: The mantissa must be ≥ 1 and < 10. If you get 45.6 × 10³, that's not proper scientific notation — the mantissa is too large. Convert to 4.56 × 10⁴.

3. Converting Small Numbers (Negative Exponent)

For numbers < 1: move the decimal point right until exactly one non-zero digit is to its left. Count the moves — the exponent is negative.

Algorithm: 1. Move the decimal point right until you pass the first non-zero digit 2. Count how many places you moved 3. The exponent is the negative of that count

Example 1: Convert 0.00072 to scientific notation.

$0.00072 = 7.2 × 10⁻⁴
         ↑
Move decimal 4 places right.
$

Example 2: Convert 0.000000401 to scientific notation.

$0.000000401 = 4.01 × 10⁻⁷
Move decimal 7 places right.
$

Example 3: Convert 0.003 to scientific notation.

$0.003 = 3 × 10⁻³
Move decimal 3 places right.
$

Key check: The exponent MUST be negative for numbers less than 1. If you write 0.0072 as 7.2 × 10³ (positive exponent), that means 7,200 — completely wrong.

4. Converting from Scientific Notation Back to Ordinary Numbers

For positive exponent (× 10ⁿ): Move the decimal point right n places, adding zeros as needed.

$3.84 × 10⁵ = 3.84000 → move 5 places right → 384,000
7.1 × 10³ = 7.100 → move 3 places right → 7,100
5 × 10⁶ = 5,000,000
$

For negative exponent (× 10⁻ⁿ): Move the decimal point left n places, adding zeros as needed.

$2.5 × 10⁻³ = 0.0025
9.03 × 10⁻⁵ = 0.0000903
1.6 × 10⁻¹ = 0.16
$

5. The Mantissa Rule: 1 ≤ a < 10

In proper scientific notation, the mantissa always satisfies 1 ≤ a < 10.

Why this matters: Standardizing the mantissa makes comparisons immediate — the number with the larger exponent is always larger (if both are positive). It also makes mental arithmetic with scientific notation consistent.

Incorrect → Correct conversions: - 0.56 × 10⁴ → 5.6 × 10³ (mantissa too small; add 1 to exponent) - 45.2 × 10⁻³ → 4.52 × 10⁻² (mantissa too large; add 1 to exponent... wait) - Actually: 45.2 × 10⁻³ = 4.52 × 10 × 10⁻³ = 4.52 × 10⁻². The exponent increased by 1 (from −3 to −2) when we divided the mantissa by 10. - 102.3 × 10² → 1.023 × 10⁴ (mantissa too large; divide by 100 means add 2 to exponent)

The balancing rule: If you divide the mantissa by 10ᵏ, add k to the exponent. If you multiply the mantissa by 10ᵏ, subtract k from the exponent.

6. Multiplying and Dividing in Scientific Notation

Multiplication: (a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10^(m+n)

1. Multiply the mantissas
2. Add the exponents
3. Adjust the mantissa to be between 1 and 10 if needed

Example 1: (3 × 10⁴) × (2 × 10³)

$= (3 × 2) × 10^(4+3)
= 6 × 10⁷
$

Example 2: (4.5 × 10⁶) × (3 × 10²)

$= (4.5 × 3) × 10^(6+2)
= 13.5 × 10⁸         ← Mantissa is too large!
= 1.35 × 10⁹         ← Divide mantissa by 10, add 1 to exponent
$

Example 3: (2.5 × 10⁻³) × (4 × 10⁷)

$= (2.5 × 4) × 10^(−3+7)
= 10 × 10⁴           ← Mantissa is too large but equals exactly 10
= 1.0 × 10⁵          ← Divide mantissa by 10, add 1 to exponent
$

Division: (a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10^(m−n)

1. Divide the mantissas
2. Subtract the exponents (m − n)
3. Adjust the mantissa if needed

Example 1: (8 × 10⁹) ÷ (2 × 10³)

$= (8 ÷ 2) × 10^(9−3)
= 4 × 10⁶
$

Example 2: (6 × 10⁸) ÷ (3 × 10⁻²)

$= (6 ÷ 3) × 10^(8−(−2))
= 2 × 10^(8+2)
= 2 × 10¹⁰
$

Example 3: (1.5 × 10⁴) ÷ (5 × 10⁶)

= (1.5 ÷ 5) × 10^(4−6)
= 0.3 × 10⁻²          ← Mantissa is too small!
= 3.0 × 10⁻³          ← Multiply mantissa by 10, subtract 1 from exponent

7. Adding and Subtracting in Scientific Notation

⚠️ THIS IS CRITICAL — Unlike multiplication and division, you CANNOT simply add or subtract exponents when adding/subtracting in scientific notation. The exponents MUST be the same first.

Algorithm: 1. Adjust one (or both) numbers so they have the SAME exponent 2. Add or subtract the mantissas 3. Keep the common exponent 4. Re-normalize the mantissa if needed

Example 1: (3.5 × 10⁶) + (2.1 × 10⁶)

$Same exponent (6), so just add mantissas:
= (3.5 + 2.1) × 10⁶
= 5.6 × 10⁶
$

Example 2: (4.2 × 10⁵) + (3.8 × 10⁴)

Different exponents. Convert both to the larger exponent (5):
3.8 × 10⁴ = 0.38 × 10⁵

Now add:
(4.2 + 0.38) × 10⁵
= 4.58 × 10⁵

Example 3: (7.0 × 10⁻³) − (2.5 × 10⁻⁴)

$Convert 2.5 × 10⁻⁴ to exponent −3:
2.5 × 10⁻⁴ = 0.25 × 10⁻³

Now subtract:
(7.0 − 0.25) × 10⁻³
= 6.75 × 10⁻³
$

Example 4 — adding very different sizes: (5.0 × 10⁸) + (3.0 × 10³)

$Convert 3.0 × 10³ to exponent 8:
3.0 × 10³ = 0.00003 × 10⁸

5.0 × 10⁸ + 0.00003 × 10⁸ ≈ 5.00003 × 10⁸
$

In practice, the small number is negligible compared to the large one — that's the power of scientific notation.

8. Comparing Magnitudes

Scientific notation makes comparing vastly different numbers trivial: just look at the exponent.

Rule: The number with the larger exponent is larger. If exponents are equal, compare mantissas.

Example: Order these from smallest to largest: - 3.6 × 10⁷ - 9.1 × 10⁵ - 2.0 × 10⁸ - 4.5 × 10⁷

Answer: 9.1 × 10⁵ < 3.6 × 10⁷ < 4.5 × 10⁷ < 2.0 × 10⁸

How many times larger? To find how many times larger A is than B: divide A by B.

Example: How many times larger is 8 × 10¹² compared to 4 × 10⁹?

$(8 × 10¹²) ÷ (4 × 10⁹) = (8 ÷ 4) × 10^(12−9) = 2 × 10³ = 2,000 times larger
$

Example: The Sun's diameter is 1.39 × 10⁹ m. Earth's diameter is 1.27 × 10⁷ m. How many times larger?

$(1.39 × 10⁹) ÷ (1.27 × 10⁷) = (1.39 ÷ 1.27) × 10^(9−7) ≈ 1.094 × 10² ≈ 109 times larger
$

9. Real-World Applications

Very large numbers: - Speed of light: 3.00 × 10⁸ m/s - Distance to nearest star (Proxima Centauri): 4.02 × 10¹³ km - Number of cells in the human body: ~3.7 × 10¹³ - Age of the universe: ~1.38 × 10¹⁰ years

Very small numbers: - Mass of an electron: 9.11 × 10⁻³¹ kg - Planck length (smallest meaningful length): 1.62 × 10⁻³⁵ m - Wavelength of visible light: ~5 × 10⁻⁷ m - One nanosecond: 1 × 10⁻⁹ s

Engineering notation (a variation): the exponent is always a multiple of 3, matching SI prefixes (kilo = 10³, mega = 10⁶, milli = 10⁻³, micro = 10⁻⁶, etc.).

Worked Examples

Example 1: Converting and Normalizing

Write 0.00000507 in scientific notation, then convert 8.04 × 10⁻⁴ to an ordinary decimal.

Solution:

Part A — 0.00000507: Move decimal right until after the first non-zero digit (5): 0.00000507 → 5.07, moved 6 places = 5.07 × 10⁻⁶

Part B — 8.04 × 10⁻⁴: Move decimal 4 places left: 8.04 → 0.000804 = 0.000804

Example 2: Multiplication and Renormalization

Calculate (4.5 × 10³) × (6.2 × 10⁸). Express in proper scientific notation.

Solution:

$(4.5 × 10³) × (6.2 × 10⁸)
= (4.5 × 6.2) × 10^(3+8)
= 27.9 × 10¹¹
$

27.9 ≥ 10, so renormalize:

$27.9 × 10¹¹ = 2.79 × 10¹² (divide mantissa by 10, add 1 to exponent)
$

Answer: 2.79 × 10¹²

Example 3: Subtraction with Different Exponents

Calculate (9.3 × 10⁷) − (4.6 × 10⁵).

Solution:

$Adjust 4.6 × 10⁵ to have exponent 7:
4.6 × 10⁵ = 0.046 × 10⁷

Now subtract:
(9.3 − 0.046) × 10⁷
= 9.254 × 10⁷
$

Since 9.254 has the same number of significant figures context as data suggests (or we can keep it as is), and 1 ≤ 9.254 < 10, it's already normalized.

Answer: 9.254 × 10⁷

Example 4: Division and Magnitude Comparison

The mass of Jupiter is 1.90 × 10²⁷ kg. The mass of Earth is 5.97 × 10²⁴ kg. How many times more massive is Jupiter than Earth?

Solution:

$Ratio = (1.90 × 10²⁷) ÷ (5.97 × 10²⁴)
      = (1.90 ÷ 5.97) × 10^(27−24)
      ≈ 0.318 × 10³
$

Renormalize: 0.318 × 10³ = 3.18 × 10² (multiply mantissa by 10, subtract 1 from exponent)

Answer: Jupiter is about 318 times more massive than Earth.

Key Terms

• larger exponent
• Converting Small Numbers (Negative Exponent)
• 267,000
• 318 times
• standard form
• Correct: D) Both A and B
• 1.0 × 10⁻¹
• Correct: B) 5.45 × 10³
• Correct: B) 3.0 × 10¹²
• Forgetting trailing zeros.
• Adding And Subtracting In Scientific Notation
• Multiplying And Dividing In Scientific Notation

Practice Problems

(Answers are below. Try each problem before checking.)

Problem 1: Convert to scientific notation: (a) 72,300,000 (b) 0.00000409

Problem 2: Convert to ordinary numbers: (a) 2.67 × 10⁵ (b) 8.1 × 10⁻⁴

Problem 3: Calculate and express in proper scientific notation: (3.0 × 10⁴) × (5.0 × 10³)

Problem 4: Calculate: (9.0 × 10⁸) ÷ (3.0 × 10²)

Problem 5: Calculate: (4.5 × 10⁶) + (3.2 × 10⁵)

Problem 6: Calculate: (7.0 × 10⁻⁴) × (2.0 × 10⁻³) ÷ (1.4 × 10⁻⁵)

Problem 7: The distance from Earth to the Sun is 1.50 × 10⁸ km. Light travels at 3.00 × 10⁵ km/s. How many seconds does light take to travel from the Sun to Earth?

Summary

1. Scientific notation writes numbers as a × 10ⁿ where 1 ≤ a < 10, making extremely large and small numbers manageable.
2. Conversion: Move the decimal to follow the first non-zero digit. Moving left → positive exponent; moving right → negative exponent.
3. Multiplication: Multiply mantissas, add exponents. Division: Divide mantissas, subtract exponents. Always renormalize.
4. Addition and subtraction require matching exponents FIRST — adjust the smaller number to match the larger exponent, then add/subtract mantissas.
5. Magnitude comparison is immediate from the exponent: bigger exponent = bigger number. The ratio tells you "how many times larger."
6. Scientific notation is essential for science, engineering, and any field dealing with extreme scales — from quantum physics to cosmology.

Pitfalls

• Writing numbers like 0.56 × 10⁴ or 45 × 10⁻³. These are not in proper scientific notation — the mantissa must be between 1 and 10.
• Forgetting to renormalize after multiplication or division. 27.9 × 10¹¹ is technically wrong for scientific notation — it should be 2.79 × 10¹².
• Adding/subtracting with different exponents without adjusting. You cannot just add 4.5 × 10⁶ and 3.2 × 10⁵ as (4.5+3.2) × 10⁶ — that's wrong.
• Using the wrong sign on the exponent. 0.004 = 4 × 10⁻³ (negative), NOT 4 × 10³ (which equals 4,000).
• Forgetting trailing zeros. 5.0 × 10³ is not the same as 5 × 10³ — the mantissa 5.0 implies 2 significant figures.

Quiz

Answer each question, then read the explanation for your choice.

Q1: Write 0.0000047 in scientific notation.

A) 47 × 10⁻⁷ B) 4.7 × 10⁻⁶ C) 4.7 × 10⁶ D) 0.47 × 10⁻⁵

Q2: What is 5.23 × 10⁴ as an ordinary number?

A) 523 B) 52,300 C) 0.000523 D) 523,000

Q3: Calculate (3.0 × 10⁵) × (4.0 × 10⁻²) in proper scientific notation.

A) 12 × 10³ B) 1.2 × 10⁴ C) 1.2 × 10² D) 12 × 10⁷

Q4: Calculate (6.0 × 10⁸) ÷ (2.0 × 10⁻⁴).

A) 3.0 × 10⁴ B) 3.0 × 10¹² C) 3.0 × 10⁻¹² D) 12 × 10⁴

Q5: Add (5.0 × 10³) + (4.5 × 10²). Write in proper scientific notation.

A) 9.5 × 10³ B) 5.45 × 10³ C) 5.45 × 10² D) 9.5 × 10⁵

Q6: Which is larger: 8.7 × 10⁻⁵ or 3.2 × 10⁻⁴?

A) 8.7 × 10⁻⁵ B) 3.2 × 10⁻⁴ C) They are equal D) Cannot be determined

Q7: The speed of light is 3.0 × 10⁸ m/s. How far does light travel in 1 nanosecond (1 × 10⁻⁹ s)?

A) 3.0 × 10⁻¹ m B) 0.3 m C) 3.0 × 10¹⁷ m D) Both A and B

Q8: Write 45,000 in proper scientific notation.

A) 45 × 10³ B) 4.5 × 10⁴ C) 0.45 × 10⁵ D) 4.5 × 10⁻⁴

Next Steps

Move on to 00-13 — Number Sequences and Patterns to learn about arithmetic and geometric sequences, the Fibonacci sequence, pattern recognition, and finding the nth term.

Q5: Which number is smallest?

A) 9.1 × 10⁻² B) 2.5 × 10⁻¹ C) 8.7 × 10⁻³ D) 1.0 × 10⁻⁴

Answer: D) 1.0 × 10⁻⁴ −4 < −3 < −2 < −1. 1.0 × 10⁻⁴ = 0.0001, which is the smallest.