Scientific Notation in Mathematics
Content Standards and Learning Competencies
Content Standards
The learners should have knowledge and understanding of operations using scientific notation. (MG)
Performance Standards
By the end of the quarter, the learners are able to write numbers in scientific notation and perform operations on numbers written in scientific notation.
Learning Competencies
At the end of the lesson, the learners are expected to:
Write numbers in scientific notation to represent very large or very small numbers, and vice versa.
Perform operations on numbers expressed in scientific notation.
What is Scientific Notation?
Definition
Scientific notation is a way of writing numbers that are too large or too small in a more concise form.
It is written as a coefficient multiplied by 10 raised to a power.
Format
A number in scientific notation is written as:
a × 10^n
Where 1 ≤ a < 10 and n is an integer
Purpose
Scientific notation makes it easier to:
Write very large numbers (like distances in space)
Write very small numbers (like sizes of atoms)
Perform calculations with extreme values
Converting to Scientific Notation
Follow these steps to convert a number to scientific notation:
Move the decimal point
Move the decimal point so that there is exactly one non-zero digit to the left of the decimal point.
Count the moves
Count how many places you moved the decimal point.
Determine the exponent
If you moved the decimal point to the left, the exponent is positive. If you moved the decimal point to the right, the exponent is negative.
Write in scientific notation
Write the number as a coefficient (between 1 and 10) multiplied by 10 raised to the power determined in step 3.
Examples: Converting from Scientific Notation
Converting from scientific notation to standard form requires moving the decimal point according to the exponent.
3.45 × 10^4
Move decimal 4 places right
Standard form: 34,500
7.891 × 10^-3
Move decimal 3 places left
Standard form: 0.007891
1.2 × 10^6
Move decimal 6 places right
Standard form: 1,200,000
9.87 × 10^-5
Move decimal 5 places left
Standard form: 0.0000987
Multiplication with Scientific Notation
Multiply Coefficients
Multiply the decimal parts (coefficients) together
Add Exponents
Add the powers of 10 (exponents) together
Express in Scientific Notation
Ensure the coefficient is between 1 and 10, adjusting the exponent if necessary
Example: (2.5 × 10^4) × (3.0 × 10^3)
Step 1: Multiply coefficients: 2.5 × 3.0 = 7.5
Step 2: Add exponents: 4 + 3 = 7
Step 3: Result: 7.5 × 10^7
Division with Scientific Notation
Dividing numbers in scientific notation follows a systematic process:
Divide Coefficients
Divide the decimal parts (coefficients)
Example: 8.4 ÷ 2.0 = 4.2
Subtract Exponents
Subtract the divisor's exponent from the dividend's exponent
Example: 6 - 2 = 4
Complete Example: (8.4 × 10^6) ÷ (2.0 × 10^2) = 4.2 × 10^4
Express in Scientific Notation
Ensure the coefficient is between 1 and 10
Final example: 4.2 × 10^4
Practice Examples
Million (10^6)
1,000,000 written in scientific notation is 1 × 10^6
Billion (10^9)
1,000,000,000 written in scientific notation is 1 × 10^9
Millionth (10^-6)
0.000001 written in scientific notation is 1 × 10^-6
Billionth (10^-9)
0.000000001 written in scientific notation is 1 × 10^-9
Real-World Applications
Scientific notation is essential in many fields where extremely large or small numbers are common:
Astronomy
Astronomers use scientific notation to express vast cosmic distances, such as the 9.46 × 10^15 meters in one light-year.
Microbiology
Microbiologists represent microscopic cell sizes and bacterial populations, often working with measurements as small as 1 × 10^-6 meters.
Computer Science
Computer scientists use scientific notation to express data storage capacities, like 2.5 × 10^12 bytes (2.5 terabytes).
Physics
Physicists rely on scientific notation for atomic and subatomic measurements, such as the 9.11 × 10^-31 kg mass of an electron.
Evaluation Activity
Activity 6: Problem 1
Daniel's computer hard disk drive holds 1.83 × 10^12 bytes of information. If he buys an extra memory stick that holds 8 × 10^9 bytes of information, how much memory will the computer hold altogether? Express your answer in decimal form and in scientific notation.
Activity 6: Problem 2
Abby is creating a mosaic in her guest room using square tiles. The width of the tile is 0.25 meters. If it took 670 tiles to cover the width of the guest room, how wide is it? Express your answer in scientific notation.
Instruction: Let the learners analyze and solve each problem. Present the rubrics of the activity to the class.
Scoring Rubrics
5 points: Complete solution with correct procedure and correct answer.
4 points: Complete solution with one incorrect procedure but correct answer.
3 points: Partially completed solution with 2-3 incorrect procedures but correct answer.
2 points: Incomplete solution with 1-2 correct procedures but incorrect answer.
1 point: Incomplete solution with an attempt but incorrect answer.
0 points: No attempt to solve the problem.
Answer Key and Learning Resources
Use these materials to check your work and deepen your understanding of scientific notation.
Answer Key
a. 1,838,000,000,000 bytes and 1.838 × 10^12 bytes
b. 1.675 × 10^2 meters
Video Resource
Dodds, C. (2012, February 6). Colin Dodds - Scientific Notation (Math Song) [Video]. YouTube.
CK-12 Foundation
CK-12 Foundation. (n.d.). CK-12 Foundation.
BYJU'S Resource
Operations with Scientific Notation (Addition, Multiplication, Subtraction of Numbers) - BYJUS. (2022, August 10).
https://byjus.com/us/math/operations-in-scientific-notation/