Writing Large and Small Numbers in Scientific Notation

Introduction

Key Concepts

Definition of Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in standard decimal form. It is particularly useful in fields such as science, engineering, and mathematics, where such numbers frequently occur. A number in scientific notation is written as:

$N = a \times 10^b$

where:

• $a$ is the coefficient, a number greater than or equal to 1 and less than 10.
• $b$ is the exponent, an integer that indicates how many places the decimal point has been moved.

For example, the number 5,600 can be written in scientific notation as:

$5.6 \times 10^3$

Converting Large Numbers to Scientific Notation

To convert a large number to scientific notation:

1. Place the decimal after the first non-zero digit.
2. Count the number of places the decimal has moved; this count becomes the exponent of 10.
3. Multiply the coefficient by 10 raised to the power of the exponent.

Example: Convert 89,000 to scientific notation.

1. Place the decimal after the first digit: 8.9

2. Count the decimal moves: 4 places.

3. Write in scientific notation: $8.9 \times 10^4$

Converting Small Numbers to Scientific Notation

To convert a small number (less than 1) to scientific notation:

1. Place the decimal after the first non-zero digit.
2. Count the number of places the decimal has moved to the right; this count becomes the negative exponent of 10.
3. Multiply the coefficient by 10 raised to the power of the negative exponent.

Example: Convert 0.0045 to scientific notation.

1. Place the decimal after the first non-zero digit: 4.5

2. Count the decimal moves: 3 places to the right.

3. Write in scientific notation: $4.5 \times 10^{-3}$

Standard Form vs. Scientific Notation

In some educational contexts, particularly in the IB MYP curriculum, scientific notation is referred to as "standard form." Both terms are often used interchangeably, although "standard form" can sometimes encompass additional formatting rules depending on the region or educational system.

Operations with Scientific Notation

Scientific notation simplifies arithmetic operations with very large or small numbers. Below are the basic operations:

Addition and Subtraction

To add or subtract numbers in scientific notation, ensure that the exponents of 10 are the same.

Example: Add $3 \times 10^4$ and $2 \times 10^4$.

$3 \times 10^4 + 2 \times 10^4 = 5 \times 10^4$

If the exponents are different, adjust the terms to have the same exponent before performing the operation.

Multiplication

To multiply numbers in scientific notation, multiply the coefficients and add the exponents.

Example: Multiply $(2 \times 10^3) \times (3 \times 10^2)$.

$2 \times 3 = 6$

$10^3 \times 10^2 = 10^{3+2} = 10^5$

Thus, $(2 \times 10^3) \times (3 \times 10^2) = 6 \times 10^5$

Division

To divide numbers in scientific notation, divide the coefficients and subtract the exponents.

Example: Divide $(5 \times 10^6) \div (2 \times 10^3)$.

$5 \div 2 = 2.5$

$10^6 \div 10^3 = 10^{6-3} = 10^3$

Thus, $(5 \times 10^6) \div (2 \times 10^3) = 2.5 \times 10^3$

Converting Scientific Notation Back to Standard Form

To convert scientific notation back to standard form:

1. Move the decimal point in the coefficient according to the exponent.
2. If the exponent is positive, move the decimal to the right.
3. If the exponent is negative, move the decimal to the left.

Example 1: Convert $7.2 \times 10^3$ to standard form.

Move the decimal 3 places to the right: 7,200

Example 2: Convert $3.5 \times 10^{-2}$ to standard form.

Move the decimal 2 places to the left: 0.035

Applications of Scientific Notation

Scientific notation is widely used in various fields to handle measurements and calculations involving extremely large or small numbers:

• Astronomy: Distances between celestial bodies, such as the distance from the Earth to the Sun, which is approximately $1.496 \times 10^{11}$ meters.
• Physics: Quantities like the charge of an electron ($-1.602 \times 10^{-19}$ coulombs).
• Chemistry: Avogadro's number, which is $6.022 \times 10^{23}$ molecules per mole.
• Engineering: Measuring data storage, e.g., terabytes and gigabytes, expressed as $10^{12}$ bytes and $10^9$ bytes respectively.
• Medicine: Doses of medication, such as nanograms ($10^{-9}$ grams) or milligrams ($10^{-3}$ grams).

Common Mistakes and How to Avoid Them

When working with scientific notation, students often make the following mistakes:

• Mistake: Incorrectly placing the decimal point.
• Solution: Always ensure that the coefficient is between 1 and 10 by placing the decimal appropriately.
• Mistake: Forgetting to adjust the exponent when converting between standard form and scientific notation.
• Solution: Carefully count the number of places the decimal has moved to determine the correct exponent.
• Mistake: Incorrectly performing arithmetic operations with the exponents.
• Solution: Remember the rules: add exponents when multiplying, subtract when dividing, and keep exponents the same when adding or subtracting.
• Mistake: Confusing positive and negative exponents.
• Solution: Positive exponents indicate large numbers (decimal moved right), while negative exponents indicate small numbers (decimal moved left).

Practice Problems

To reinforce the understanding of scientific notation, here are some practice problems:

1. Convert the following numbers to scientific notation:
   • a) 0.00056
   • b) 78,900
2. Perform the following operations:
   • a) $(4 \times 10^5) + (3 \times 10^5)$
   • b) $(6 \times 10^{-4}) \times (2 \times 10^3)$
3. Convert the scientific notation back to standard form:
   • a) $9.1 \times 10^2$
   • b) $5.3 \times 10^{-6}$

Answers:

• a) $5.6 \times 10^{-4}$
• b) $7.89 \times 10^4$
• a) $7 \times 10^5$
• b) $1.2 \times 10^{-1}$
• a) 910
• b) 0.0000053

Comparison Table

Summary and Key Takeaways