Standard to Scientific Form
Converting a number from standard form to scientific notation means moving the decimal point to just after the first non-zero digit, counting the number of places moved, and writing the result as a decimal between 1 and 10 times a power of 10. For 81,200,000: move the decimal 7 places left to get 8.12, so it is 8.12 times 10 to the 7. This Grade 7 math skill from Saxon Math, Course 2 allows extremely large numbers to be expressed compactly and is fundamental for science, engineering, and any quantitative field dealing with measurements that span many orders of magnitude.
Key Concepts
Property To write a number in scientific notation, place the decimal point to the right of the first nonzero digit. Then, count the number of places the decimal moved from its original position to determine the exponent for the power of 10.
Examples For $81,000,000$: place the decimal after 8 to get $8.1$. The decimal hopped 7 places, so it's $8.1 \times 10^7$. For $602,500,000$: place the decimal after 6 to get $6.025$. The decimal hopped 8 places, so it's $6.025 \times 10^8$. A football field is $110,000$ mm, which becomes $1.1 \times 10^5$ mm after the decimal hops 5 places.
Explanation Think of it as pinning the decimal right after the first important digit. Then, count how many hops the decimal made from where it started to its new home. That number of hops becomes your exponent for the power of 10. This trick lets you neatly package huge numbers without writing all those trailing zeros.
Common Questions
How do I convert a large number to scientific notation?
Place the decimal after the first non-zero digit to get a number between 1 and 10. Count how many places the decimal moved — that is the positive exponent. For 81,200,000: 8.12 times 10 to the 7.
What must the coefficient in scientific notation be?
The coefficient (the decimal part) must be at least 1 but less than 10. If it is not, move the decimal and adjust the exponent accordingly.
How do I convert a small decimal to scientific notation?
Move the decimal right until you have one non-zero digit before it. Count the moves — that is the absolute value of the negative exponent. For 0.000071: 7.1 times 10 to the -5.
What is the exponent if the decimal moves 4 places to the left?
Moving the decimal 4 places to the left means the original number was large, so the exponent is positive 4. The number is in the form (something) times 10 to the 4.
When do students learn to convert to scientific notation?
Scientific notation conversion is a Grade 7 skill. Saxon Math, Course 2 covers it in Chapter 9 as both a reading and writing skill.
Why do scientists use scientific notation instead of standard form?
Writing 602,000,000,000,000,000,000,000 (Avogadro's number) is impractical. Scientific notation (6.02 times 10 to the 23) is compact, precise, and easy to use in calculations.
What are common mistakes when converting to scientific notation?
Students sometimes forget the exponent tells how many places the decimal moved, not how many zeros the number has. Counting places of movement gives the correct exponent.