High School Chemistry: Measurement and Safety - Scientific Notation - Writing Large and Small Numbers

Scientific Notation: Writing Large and Small Numbers

1. What This Is (In Plain English)

Scientific notation is a way to write really big or really small numbers in a shorter, more manageable form.

Why does it matter? Without scientific notation, we wouldn't have calculators that can handle huge numbers like the number of stars in the universe (about 100,000,000,000,000,000,000). We also wouldn't be able to write about tiny things like atoms and molecules, which are too small to count.

2. Key Ideas & Definitions

• Scientific Notation: A way to write numbers in the form a × 10^n, where a is a number between 1 and 10, and n is an integer.
  • Example: 456,000,000 = 4.56 × 10^8
  • Memory trick: Think of it like a big number divided by a power of 10. For example, 456,000,000 ÷ 100,000,000 = 4.56.
• Coefficient: The number a in scientific notation.
  • Example: In 4.56 × 10^8, the coefficient is 4.56.
  • Memory trick: Think of it like a multiplier. For example, 4.56 × 10^8 means 4.56 multiplied by 10 to the power of 8.
• Exponent: The number n in scientific notation.
  • Example: In 4.56 × 10^8, the exponent is 8.
  • Memory trick: Think of it like a power of 10. For example, 10^8 means 10 multiplied by itself 8 times.
• Significant Figures: The number of digits in the coefficient that are known to be accurate.
  • Example: In 4.56 × 10^8, there are 4 significant figures.
  • Memory trick: Think of it like the number of digits you're sure about. For example, if you measured something to be 4.56 meters, you're sure about the first 4 digits.
• Standard Form: A number written in its standard form, without scientific notation.
  • Example: 456,000,000 in standard form is 456,000,000.
  • Memory trick: Think of it like the normal way we write numbers. For example, 456,000,000 is just a big number written normally.
• Expanded Form: A number written in its expanded form, with powers of 10.
  • Example: 456,000,000 in expanded form is 4.56 × 10^8.
  • Memory trick: Think of it like a big number broken down into smaller parts. For example, 456,000,000 = 4.56 × 100,000,000.
• Simplifying Scientific Notation: Writing a number in scientific notation with the fewest possible digits.
  • Example: 456,000,000 can be simplified to 4.56 × 10^8.
  • Memory trick: Think of it like finding the simplest way to write a number. For example, 456,000,000 can be simplified by dividing it by 100,000,000.
• Converting Between Scientific Notation and Standard Form: Writing a number in either scientific notation or standard form.
  • Example: 4.56 × 10^8 can be converted to standard form as 456,000,000.
  • Memory trick: Think of it like switching between two different ways of writing numbers. For example, you can switch between scientific notation and standard form by multiplying or dividing by powers of 10.

3. How To Do It (Step-by-Step)

1. Write a number in scientific notation:
   • Take a big or small number and divide it by a power of 10 to get a number between 1 and 10.
   • Write the number between 1 and 10 as the coefficient.
   • Write the power of 10 as the exponent.
   • Example: Write 456,000,000 in scientific notation.
     • Divide 456,000,000 by 100,000,000 to get 4.56.
     • Write 4.56 as the coefficient.
     • Write 8 as the exponent.
     • The answer is 4.56 × 10^8.
2. Simplify a number in scientific notation:
   • Take a number in scientific notation and divide the coefficient by a power of 10 to get a number between 1 and 10.
   • Write the new coefficient.
   • Write the new exponent.
   • Example: Simplify 4.56 × 10^9.
     • Divide 4.56 by 10 to get 0.456.
     • Write 0.456 as the new coefficient.
     • Write 9 - 1 = 8 as the new exponent.
     • The answer is 4.56 × 10^8.
3. Convert a number from scientific notation to standard form:
   • Take a number in scientific notation and multiply the coefficient by a power of 10 to get the original number.
   • Write the original number.
   • Example: Convert 4.56 × 10^8 to standard form.
     • Multiply 4.56 by 10^8 to get 456,000,000.
     • Write 456,000,000 as the original number.
4. Convert a number from standard form to scientific notation:
   • Take a number in standard form and divide it by a power of 10 to get a number between 1 and 10.
   • Write the number between 1 and 10 as the coefficient.
   • Write the power of 10 as the exponent.
   • Example: Convert 456,000,000 to scientific notation.
     • Divide 456,000,000 by 100,000,000 to get 4.56.
     • Write 4.56 as the coefficient.
     • Write 8 as the exponent.
     • The answer is 4.56 × 10^8.

4. Watch Out! (Common Mistakes)

• Mistake: Writing a number in scientific notation with a coefficient that's not between 1 and 10.
  • Fix: Divide the coefficient by a power of 10 to get a number between 1 and 10.
  • Example: Write 456,000,000 in scientific notation.
    • Divide 456,000,000 by 100,000,000 to get 4.56.
    • Write 4.56 as the coefficient.
    • Write 8 as the exponent.
    • The answer is 4.56 × 10^8.
• Mistake: Writing a number in scientific notation with an exponent that's not an integer.
  • Fix: Write the exponent as an integer.
  • Example: Write 456,000,000 in scientific notation.
    • Divide 456,000,000 by 100,000,000 to get 4.56.
    • Write 4.56 as the coefficient.
    • Write 8 as the exponent.
    • The answer is 4.56 × 10^8.
• Mistake: Converting a number from scientific notation to standard form incorrectly.
  • Fix: Multiply the coefficient by a power of 10 to get the original number.
  • Example: Convert 4.56 × 10^8 to standard form.
    • Multiply 4.56 by 10^8 to get 456,000,000.
    • Write 456,000,000 as the original number.

5. Practice Problems

Problem 1: Write 456,000,000 in scientific notation.

Solution: Divide 456,000,000 by 100,000,000 to get 4.56. Write 4.56 as the coefficient. Write 8 as the exponent. The answer is 4.56 × 10^8.

Problem 2: Simplify 4.56 × 10^9.

Solution: Divide 4.56 by 10 to get 0.456. Write 0.456 as the new coefficient. Write 9 - 1 = 8 as the new exponent. The answer is 4.56 × 10^8.

6. Cram Sheet

• Scientific Notation: A way to write numbers in the form a × 10^n, where a is a number between 1 and 10, and n is an integer.
• Coefficient: The number a in scientific notation.
• Exponent: The number n in scientific notation.
• Significant Figures: The number of digits in the coefficient that are known to be accurate.
• Standard Form: A number written in its standard form, without scientific notation.
• Expanded Form: A number written in its expanded form, with powers of 10.
• Simplifying Scientific Notation: Writing a number in scientific notation with the fewest possible digits.
• Converting Between Scientific Notation and Standard Form: Writing a number in either scientific notation or standard form.
• Mass stays the same during a phase change; energy is what changes.

7. Where to Learn More

• YouTube: The Amoeba Sisters have a great video on scientific notation.
• PhET Simulation: The PhET simulation on scientific notation is a great interactive tool.
• School-Friendly Website: The Khan Academy website has a great section on scientific notation.