Basic Math

Scientific notation | Ninth Grade

Scientific Notation - Ninth Grade Math

Introduction to Scientific Notation

Scientific Notation: A method of writing very large or very small numbers using powers of 10
Purpose: Makes it easier to work with extremely large or small numbers
Used in: Science, astronomy, chemistry, physics, engineering, and technology
General Form of Scientific Notation:
$$a \times 10^n$$

where:
• $a$ = coefficient (a number where $1 \leq |a| < 10$)
• $10$ = base (always 10)
• $n$ = exponent (any integer: positive, negative, or zero)

Requirements for coefficient $a$:
• Must be greater than or equal to 1
• Must be less than 10
• Only ONE non-zero digit before the decimal point
Key Rules:
Large numbers: Use positive exponent
  Example: $3,500,000 = 3.5 \times 10^6$

Small numbers (decimals): Use negative exponent
  Example: $0.0025 = 2.5 \times 10^{-3}$

Numbers between 1 and 10: Exponent is 0
  Example: $5 = 5 \times 10^0$

1. Convert Between Standard and Scientific Notation

Standard to Scientific Notation

Standard Form: The regular way of writing numbers (e.g., 450,000)
Scientific Notation: Numbers written as $a \times 10^n$
Steps to Convert Standard to Scientific Notation:
Step 1: Place decimal point after the first non-zero digit
Step 2: Count how many places you moved the decimal
Step 3: If original number $\geq 10$: exponent is positive
          If original number $< 1$: exponent is negative
Step 4: Write as $a \times 10^n$
Step 5: Check: $1 \leq a < 10$
Example 1: Convert 6,500,000 to scientific notation

Step 1: Move decimal after first digit: $6.5$
Step 2: Counted 6 places to the left
Step 3: Original number is large → positive exponent
Answer: $6.5 \times 10^6$
Example 2: Convert 0.000043 to scientific notation

Step 1: Move decimal after first non-zero digit: $4.3$
Step 2: Counted 5 places to the right
Step 3: Original number is small → negative exponent
Answer: $4.3 \times 10^{-5}$
Example 3: Convert 892,000,000 to scientific notation

$892,000,000 = 8.92 \times 10^8$
(Moved decimal 8 places left)
Example 4: Convert 0.00709 to scientific notation

$0.00709 = 7.09 \times 10^{-3}$
(Moved decimal 3 places right)

Scientific to Standard Notation

Steps to Convert Scientific to Standard Notation:
Step 1: Identify the exponent $n$
Step 2: If $n$ is positive: Move decimal $n$ places to the RIGHT
          If $n$ is negative: Move decimal $|n|$ places to the LEFT
Step 3: Fill empty spaces with zeros
Step 4: Write the final number
Example 1: Convert $3.5 \times 10^4$ to standard form

Exponent is 4 (positive): Move decimal 4 places right
$3.5 \rightarrow 35 \rightarrow 350 \rightarrow 3500 \rightarrow 35000$
Answer: 35,000
Example 2: Convert $7.2 \times 10^{-3}$ to standard form

Exponent is -3 (negative): Move decimal 3 places left
$7.2 \rightarrow 0.72 \rightarrow 0.072 \rightarrow 0.0072$
Answer: 0.0072
Example 3: Convert $9.81 \times 10^2$ to standard form

$9.81 \times 10^2 = 981$
Answer: 981
Example 4: Convert $4.56 \times 10^{-4}$ to standard form

$4.56 \times 10^{-4} = 0.000456$
Answer: 0.000456
Quick Tips:
• Positive exponent → number gets LARGER (move decimal right)
• Negative exponent → number gets SMALLER (move decimal left)
• Exponent tells you how many places to move the decimal
• Always check: does your answer make sense?

2. Compare Numbers Written in Scientific Notation

Comparing: Determining which number is larger or smaller
Steps to Compare Numbers in Scientific Notation:
Step 1: Compare the exponents first
    • Larger exponent = larger number (if both positive)
    • Less negative exponent = larger number (if both negative)
Step 2: If exponents are equal, compare the coefficients
Step 3: Use $<$, $>$, or $=$ to show relationship
Comparison Rules:

1. Both Exponents Positive:
• Larger exponent → larger number
• If exponents equal → larger coefficient → larger number

2. Both Exponents Negative:
• Less negative exponent → larger number
• Example: $10^{-2} > 10^{-5}$ (because $-2 > -5$)

3. One Positive, One Negative:
• Positive exponent → larger number
• Example: $10^3 > 10^{-3}$
Example 1: Compare $3.2 \times 10^5$ and $4.1 \times 10^3$

Compare exponents: $5 > 3$
Therefore: $3.2 \times 10^5 > 4.1 \times 10^3$

Verification:
$3.2 \times 10^5 = 320,000$
$4.1 \times 10^3 = 4,100$
$320,000 > 4,100$ ✓
Example 2: Compare $5.6 \times 10^4$ and $8.2 \times 10^4$

Exponents are equal: Compare coefficients
$8.2 > 5.6$
Therefore: $8.2 \times 10^4 > 5.6 \times 10^4$
Example 3: Compare $2.5 \times 10^{-3}$ and $7.1 \times 10^{-5}$

Both exponents negative: $-3 > -5$
Therefore: $2.5 \times 10^{-3} > 7.1 \times 10^{-5}$

Verification:
$2.5 \times 10^{-3} = 0.0025$
$7.1 \times 10^{-5} = 0.000071$
$0.0025 > 0.000071$ ✓
Example 4: Order from smallest to largest:
$6.2 \times 10^{-2}$, $3.4 \times 10^5$, $1.9 \times 10^{-2}$, $8.7 \times 10^3$

Group by exponent:
• Negative: $6.2 \times 10^{-2}$ and $1.9 \times 10^{-2}$
• Positive: $3.4 \times 10^5$ and $8.7 \times 10^3$

Order:
$1.9 \times 10^{-2} < 6.2 \times 10^{-2} < 8.7 \times 10^3 < 3.4 \times 10^5$

3. Add and Subtract Numbers in Scientific Notation

Key Requirement: Exponents must be the SAME before adding or subtracting
Addition/Subtraction Rule:
$$(a \times 10^n) \pm (b \times 10^n) = (a \pm b) \times 10^n$$

If exponents are the same:
• Add or subtract the coefficients
• Keep the exponent the same

If exponents are different:
• Adjust one number so exponents match
• Then add or subtract

Same Exponents

Example 1: Add $(3.2 \times 10^5) + (4.7 \times 10^5)$

Exponents are the same (both $10^5$):
$(3.2 + 4.7) \times 10^5 = 7.9 \times 10^5$
Answer: $7.9 \times 10^5$
Example 2: Subtract $(8.5 \times 10^{-3}) - (2.3 \times 10^{-3})$

$(8.5 - 2.3) \times 10^{-3} = 6.2 \times 10^{-3}$
Answer: $6.2 \times 10^{-3}$

Different Exponents

When Exponents are Different:
Step 1: Choose which exponent to keep (usually the larger one)
Step 2: Adjust the other number to match that exponent
    • Move decimal and adjust exponent accordingly
Step 3: Add or subtract the coefficients
Step 4: Keep the exponent
Step 5: Adjust final answer to proper scientific notation if needed
Example 3: Add $(4.5 \times 10^6) + (3.2 \times 10^5)$

Exponents different: 6 and 5
Adjust second number to match $10^6$:
$3.2 \times 10^5 = 0.32 \times 10^6$

Now add:
$(4.5 \times 10^6) + (0.32 \times 10^6) = (4.5 + 0.32) \times 10^6 = 4.82 \times 10^6$
Answer: $4.82 \times 10^6$
Example 4: Subtract $(7.8 \times 10^4) - (5.2 \times 10^3)$

Convert $5.2 \times 10^3$ to $10^4$ form:
$5.2 \times 10^3 = 0.52 \times 10^4$

$(7.8 \times 10^4) - (0.52 \times 10^4) = (7.8 - 0.52) \times 10^4 = 7.28 \times 10^4$
Answer: $7.28 \times 10^4$
Example 5: Add $(2.5 \times 10^{-2}) + (3.4 \times 10^{-3})$

Convert $3.4 \times 10^{-3}$ to $10^{-2}$ form:
$3.4 \times 10^{-3} = 0.34 \times 10^{-2}$

$(2.5 + 0.34) \times 10^{-2} = 2.84 \times 10^{-2}$
Answer: $2.84 \times 10^{-2}$
Adjusting After Operations:
If result doesn't follow form $1 \leq a < 10$, adjust:

Example: $(9.2 + 5.3) \times 10^4 = 14.5 \times 10^4$
Since $14.5 > 10$, adjust:
$14.5 \times 10^4 = 1.45 \times 10^5$

4. Multiply Numbers in Scientific Notation

Multiplication Rule: Multiply coefficients and ADD exponents
Multiplication Formula:
$$(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$$

Steps:
1. Multiply the coefficients: $a \times b$
2. Add the exponents: $m + n$
3. Combine: $(a \times b) \times 10^{m+n}$
4. Adjust to proper scientific notation if needed
Example 1: Multiply $(3 \times 10^4) \times (2 \times 10^3)$

Multiply coefficients: $3 \times 2 = 6$
Add exponents: $4 + 3 = 7$
Answer: $6 \times 10^7$
Example 2: Multiply $(4.5 \times 10^6) \times (2.0 \times 10^{-3})$

Multiply coefficients: $4.5 \times 2.0 = 9.0$
Add exponents: $6 + (-3) = 3$
Answer: $9.0 \times 10^3$ or $9 \times 10^3$
Example 3: Multiply $(5.2 \times 10^{-4}) \times (3.0 \times 10^{-5})$

Multiply: $5.2 \times 3.0 = 15.6$
Add exponents: $-4 + (-5) = -9$
Result: $15.6 \times 10^{-9}$

Adjust (15.6 > 10):
$15.6 \times 10^{-9} = 1.56 \times 10^{-8}$
Answer: $1.56 \times 10^{-8}$
Example 4: Multiply $(2.5 \times 10^7) \times (4.0 \times 10^2)$

$(2.5 \times 4.0) \times 10^{7+2} = 10.0 \times 10^9$

Adjust: $10.0 \times 10^9 = 1.0 \times 10^{10}$
Answer: $1 \times 10^{10}$
Example 5: Multiply $(6.5 \times 10^{-2}) \times (1.2 \times 10^5)$

$(6.5 \times 1.2) \times 10^{-2+5} = 7.8 \times 10^3$
Answer: $7.8 \times 10^3$

5. Divide Numbers in Scientific Notation

Division Rule: Divide coefficients and SUBTRACT exponents
Division Formula:
$$\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$$

Steps:
1. Divide the coefficients: $\frac{a}{b}$
2. Subtract the exponents: $m - n$
3. Combine: $\left(\frac{a}{b}\right) \times 10^{m-n}$
4. Adjust to proper scientific notation if needed
Example 1: Divide $\frac{8 \times 10^7}{2 \times 10^3}$

Divide coefficients: $\frac{8}{2} = 4$
Subtract exponents: $7 - 3 = 4$
Answer: $4 \times 10^4$
Example 2: Divide $\frac{6.4 \times 10^5}{3.2 \times 10^2}$

Divide: $\frac{6.4}{3.2} = 2$
Subtract: $5 - 2 = 3$
Answer: $2 \times 10^3$
Example 3: Divide $\frac{9.6 \times 10^{-3}}{4.8 \times 10^{-7}}$

Divide: $\frac{9.6}{4.8} = 2$
Subtract: $-3 - (-7) = -3 + 7 = 4$
Answer: $2 \times 10^4$
Example 4: Divide $\frac{7.5 \times 10^4}{3.0 \times 10^6}$

Divide: $\frac{7.5}{3.0} = 2.5$
Subtract: $4 - 6 = -2$
Answer: $2.5 \times 10^{-2}$
Example 5: Divide $\frac{5.4 \times 10^{-2}}{9.0 \times 10^{-5}}$

Divide: $\frac{5.4}{9.0} = 0.6$
Subtract: $-2 - (-5) = 3$
Result: $0.6 \times 10^3$

Adjust (0.6 < 1):
$0.6 \times 10^3 = 6 \times 10^2$
Answer: $6 \times 10^2$

Operations Summary Table

OperationRuleExample
AdditionSame exponent: $(a \times 10^n) + (b \times 10^n) = (a + b) \times 10^n$$(3 \times 10^4) + (2 \times 10^4) = 5 \times 10^4$
SubtractionSame exponent: $(a \times 10^n) - (b \times 10^n) = (a - b) \times 10^n$$(7 \times 10^3) - (4 \times 10^3) = 3 \times 10^3$
MultiplicationMultiply coefficients, add exponents: $(a \times 10^m)(b \times 10^n) = ab \times 10^{m+n}$$(2 \times 10^3)(3 \times 10^4) = 6 \times 10^7$
DivisionDivide coefficients, subtract exponents: $\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$$\frac{8 \times 10^6}{2 \times 10^2} = 4 \times 10^4$

Real-World Applications

Where Scientific Notation is Used:

Astronomy:
• Distance from Earth to Sun: $1.496 \times 10^8$ km
• Speed of light: $3.0 \times 10^8$ m/s

Biology:
• Size of bacteria: $5 \times 10^{-7}$ meters
• Number of cells in human body: $3.7 \times 10^{13}$

Chemistry:
• Avogadro's number: $6.022 \times 10^{23}$ molecules/mol
• Mass of electron: $9.11 \times 10^{-31}$ kg

Technology:
• Computer storage: $1$ TB = $1 \times 10^{12}$ bytes
• Nanotechnology: $1$ nanometer = $1 \times 10^{-9}$ meters

Quick Conversion Reference

Standard FormScientific NotationWord Form
1,000,000$1 \times 10^6$One million
100,000$1 \times 10^5$One hundred thousand
10,000$1 \times 10^4$Ten thousand
1,000$1 \times 10^3$One thousand
100$1 \times 10^2$One hundred
10$1 \times 10^1$Ten
1$1 \times 10^0$One
0.1$1 \times 10^{-1}$One tenth
0.01$1 \times 10^{-2}$One hundredth
0.001$1 \times 10^{-3}$One thousandth
0.0001$1 \times 10^{-4}$One ten-thousandth

Common Mistakes to Avoid

❌ WRONG:
• Writing $45 \times 10^3$ (coefficient must be between 1 and 10!)
  ✓ CORRECT: $4.5 \times 10^4$

• Writing $0.5 \times 10^6$ (coefficient must be $\geq 1$!)
  ✓ CORRECT: $5 \times 10^5$

• Multiplying: $(2 \times 10^3) \times (3 \times 10^4) = 6 \times 10^{12}$ ✗
  ✓ CORRECT: Add exponents → $6 \times 10^7$

• Dividing: $\frac{8 \times 10^6}{2 \times 10^2} = 4 \times 10^3$ ✗
  ✓ CORRECT: Subtract exponents → $4 \times 10^4$

• Adding with different exponents directly:
  $(3 \times 10^5) + (2 \times 10^3) = 5 \times 10^8$ ✗
  ✓ CORRECT: Must make exponents same first!

• Moving decimal wrong direction for negative exponent

Step-by-Step Problem-Solving Guide

General Strategy:

For Conversions:
1. Identify if converting TO or FROM scientific notation
2. Find where decimal should go (between 1 and 10)
3. Count places moved
4. Determine sign of exponent

For Operations:
1. Identify the operation
2. Check if exponents need adjusting (add/subtract only)
3. Apply the appropriate rule
4. Simplify
5. Adjust final answer to proper form if needed

Always Check:
• Is coefficient between 1 and 10?
• Does the answer make sense?
• Did you use the correct operation on exponents?
Success Tips for Scientific Notation:
✓ Remember: coefficient must be $1 \leq a < 10$
✓ Positive exponent = large number; Negative exponent = small number
✓ Addition/Subtraction: exponents must match!
✓ Multiplication: multiply coefficients, ADD exponents
✓ Division: divide coefficients, SUBTRACT exponents
✓ Move decimal right for positive exponent, left for negative
✓ Count carefully when moving decimals
✓ Always adjust final answer to proper scientific notation
✓ Use calculator for complex coefficient calculations
✓ Practice converting back and forth for accuracy checks
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