Scientific Notation
Scientific notation writes a number as a number from 1 up to 10, times a power of 10. It keeps huge and tiny numbers short.
What you'll be able to do
- Write large and small numbers in scientific notation.
- Change them back to standard form.
- Compare and order numbers in scientific notation.
- Estimate how many times as large one number is.
The big idea
The form. A number in scientific notation looks like 3.84 × 10⁵. The first part is at least 1 and less than 10. The exponent tells how far the decimal point moved.
Large numbers. Move the decimal point left until one digit is in front of it. Count the moves. So 384,000 = 3.84 × 10⁵.
Small numbers. Move the decimal point right until one nonzero digit is in front. Count the moves and make the exponent negative. So 0.0062 = 6.2 × 10⁻³.
Back to standard form. A positive exponent moves the point right. A negative exponent moves it left. So 4.1 × 10³ = 4,100, and 4.1 × 10⁻³ = 0.0041.
Comparing. Look at the exponents first. 3 × 10⁸ is bigger than 9 × 10⁷, because 10⁸ is ten times 10⁷. If the exponents match, compare the first parts.
How many times as large. Divide the first parts and subtract the exponents. (8 × 10⁹) ÷ (2 × 10⁵) = 4 × 10⁴, which is 40,000.
Fixing other forms. 45 × 10³ is not in scientific notation, because 45 is 10 or more. 45 = 4.5 × 10¹, so 45 × 10³ = 4.5 × 10⁴.
Words you need
- Scientific notation
- A number from 1 up to 10, times a power of 10. Like 3.84 × 10⁵.
- Standard form
- The usual way to write a number, like 384,000 or 0.0062.
- Exponent
- The small raised number. In 10⁵, the exponent is 5.
- Power of 10
- 10 multiplied by itself. 10³ = 1,000 and 10⁻² = 1/100.
Reading the exponent
| Standard form | Scientific notation | Decimal point moves |
|---|---|---|
| 5,200,000 | 5.2 × 10⁶ | 6 left |
| 4,000 | 4 × 10³ | 3 left |
| 0.03 | 3 × 10⁻² | 2 right |
| 0.000105 | 1.05 × 10⁻⁴ | 4 right |
Worked examples
Example 1
The Sun is about 93,000,000 miles from Earth. Write that in scientific notation.
- Move the decimal point left until one digit, 9, is in front: 9.3.
- That took 7 moves. So the answer is 9.3 × 10⁷ miles.
Answer: 9.3 × 10⁷
Example 2
Write 0.00048 in scientific notation.
- Move the decimal point right until one nonzero digit is in front: 4.8.
- That took 4 moves to the right, so the exponent is −4.
- The answer is 4.8 × 10⁻⁴.
Answer: 4.8 × 10⁻⁴
Example 3
Write 4.05 × 10⁶ in standard form.
- A positive exponent means a large number. Move the point 6 places right.
- 4.05 becomes 4,050,000.
Answer: 4,050,000
Common mistakes
Leaving the first part too big: 45 × 10³. The first part must be less than 10. 45 × 10³ = 4.5 × 10⁴.
Using a positive exponent for a small number: 0.007 = 7 × 10³. A number below 1 has a negative exponent. 0.007 = 7 × 10⁻³.
Comparing only the first parts: 9 × 10⁴ is bigger than 2 × 10⁶. Compare the exponents first. 2 × 10⁶ = 2,000,000 is much bigger.
Counting zeros: 3.2 × 10⁵ = 3,200,000. Count places, not zeros. Move the point 5 places: 320,000.
Practice: 20 questions
13 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Light travels about 299,800 kilometers each second. Write that in scientific notation. Word problem
- A dust mite is about 0.00025 meters long. Write that in scientific notation. Word problem
- Write 7.06 × 10⁵ in standard form.
- A grain of pollen is about 3.5 × 10⁻⁴ meters wide. Write that in standard form. Word problem
- Ana writes 45,000 as 45 × 10³. Find her mistake. What is the correct form?
Tier 2 Intermediate One more step to think through
- Which is greater, 9 × 10⁴ or 2 × 10⁶?
- Earth has about 8 × 10⁹ people. A large city has about 4 × 10⁶. How many times as many people live on Earth? Word problem
- About 4,876,000,000 people are online. Estimate that as one digit times a power of 10. Word problem
- A virus is 3.1 × 10⁻⁵ m wide. A bacterium is 9.8 × 10⁻⁶ m wide. A cell is 1.2 × 10⁻⁴ m wide. Order them from smallest to largest. Word problem
- Write 0.62 × 10⁻³ in scientific notation.
Tier 3 Advanced Several steps - find the hidden structure
- A grain of sand is about 5 × 10⁻⁴ meters wide. A virus is about 1 × 10⁻⁷ meters wide. How many times as wide is the sand? Word problem
- Mount Everest is about 8.85 × 10³ meters tall. 1 meter is 10³ millimeters. How tall is it in millimeters, in scientific notation? Word problem
- A plant cell is 0.00012 meters wide. A pollen grain is 3 × 10⁻⁵ meters wide. How many times as wide is the cell? Word problem
- The Sun's mass is about 2 × 10³⁰ kg. Earth's mass is about 6 × 10²⁴ kg. About how many times as massive is the Sun? Give one digit times a power of 10. Word problem
- Leo rewrites 0.045 × 10⁻³ in scientific notation as 4.5 × 10⁻¹. Find his mistake. What is the correct form?
Tier 4 Challenge Stretch problems, some above grade level
- Put 4 × 10⁻³, 0.041 × 10⁻¹ and 380 × 10⁻⁵ in order from least to greatest.
- Which is greater, 5 × 10²⁰ + 5 × 10²⁰ or 10²¹?
- There are about 3.15 × 10⁷ seconds in a year. How many seconds has a 10-year-old been alive? Write your answer in scientific notation. Word problem
- A googol is 10¹⁰⁰. Divide it by 2. Write the answer in scientific notation. Word problem
- Significant figures are the digits from the first nonzero digit on the left. For example, 42,680 rounded to 3 significant figures is 42,700. The world population was 8,118,835,999 on one day. Round it to 3 significant figures and write it in scientific notation. Word problem Above grade level
Answers
Tier 1 Foundations
- 2.998 × 10⁵. Move the decimal point 5 places left, so one nonzero digit is in front of it.
- 2.5 × 10⁻⁴. Move the decimal point 4 places right, so one nonzero digit is in front of it.
- 706,000. Move the decimal point 5 places right.
- 0.00035. Move the decimal point 4 places left.
- 4.5 × 10⁴. Ana's first part, 45, is 10 or more. 45 = 4.5 × 10¹. So the exponent changes by 1: 3 + 1 = 4. So 45 × 10³ = 4.5 × 10⁴.
Tier 2 Intermediate
- 2 × 10⁶ is greater. Compare the exponents first: 6 is greater than 4.
- 2,000. Divide: (8 × 10⁹) ÷ (4 × 10⁶) = 2,000.
- about 5 × 10⁹. 4,876,000,000 is about 4.9 × 10⁹. To one digit, that rounds to 5 × 10⁹.
- 9.8 × 10⁻⁶ < 3.1 × 10⁻⁵ < 1.2 × 10⁻⁴. Compare the exponents first: −6 < −5 < −4.
- 6.2 × 10⁻⁴. 0.62 = 6.2 × 10⁻¹. So the exponent changes by −1: −3 + (−1) = −4. So 0.62 × 10⁻³ = 6.2 × 10⁻⁴.
Tier 3 Advanced
- 5,000. Divide: (5 × 10⁻⁴) ÷ (1 × 10⁻⁷) = 5,000.
- 8.85 × 10⁶. 8.85 × 1 = 8.85, and 10³ × 10³ = 10⁶ (add the exponents).
- 4. 0.00012 = 1.2 × 10⁻⁴. Divide: 1.2 ÷ 3 = 0.4, and 10⁻⁴ ÷ 10⁻⁵ = 10¹. So 0.4 × 10 = 4.
- about 3 × 10⁵. 2 ÷ 6 is about 0.33, and 10³⁰ ÷ 10²⁴ = 10⁶. So the ratio is about 0.33 × 10⁶, or about 3 × 10⁵.
- 4.5 × 10⁻⁵. Leo changed the exponent the wrong way. Making the first part bigger makes the exponent smaller. 0.045 = 4.5 × 10⁻². So the exponent changes by −2: −3 + (−2) = −5. So 0.045 × 10⁻³ = 4.5 × 10⁻⁵.
Tier 4 Challenge
- 380 × 10⁻⁵ < 4 × 10⁻³ < 0.041 × 10⁻¹. Rewrite each one first. 0.041 × 10⁻¹ = 4.1 × 10⁻³. 380 × 10⁻⁵ = 3.8 × 10⁻³. Now the exponents match, so compare 3.8, 4 and 4.1.
- They are equal. Two copies of 5 × 10²⁰ make 10 × 10²⁰. And 10 × 10²⁰ = 10²¹. So they are equal.
- 3.15 × 10⁸. 10 years is 10 × 3.15 × 10⁷ seconds. Multiplying by 10 adds 1 to the exponent. So it is 3.15 × 10⁸ seconds.
- 5 × 10⁹⁹. 10¹⁰⁰ = 10 × 10⁹⁹. Half of 10 is 5. So half a googol is 5 × 10⁹⁹, not 10⁵⁰.
- about 8.12 × 10⁹ people. The first three significant figures are 8, 1 and 1. The next digit, 8, rounds the last 1 up to 2. So the answer is 8.12 × 10⁹.
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