English:Scientific Notation
Scientific Notation
Introduction
Scientific notation is a compact way to write very large and very small numbers by using powers of ten. Instead of writing a long row of zeros, you write a coefficient and an exponent. For example, 5,300,000 can be written as 5.3 × 106, and 0.000042 can be written as 4.2 × 10-5.
You meet numbers on very different scales in mathematics, science, astronomy, chemistry, physics, computing, and engineering. Scientific notation makes these numbers easier to read, compare, and calculate with.
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The image above compares objects across many orders of magnitude. Each step of one power of ten changes scale by a factor of 10.
This Math Antics video introduces scientific notation and shows why powers of ten are useful.
Learning Goals
By the end of this aiMOOC, you should be able to explain scientific notation, convert between decimal notation and scientific notation, interpret positive and negative exponents, compare numbers written with powers of ten, and carry out basic calculations using scientific notation. You should also be able to decide whether a scientific-notation answer is reasonable in a real-world context.
Why Scientific Notation Is Useful
Large and small numbers can be difficult to read when they contain many zeros. Scientific notation shows the important digits separately from the scale of the number. The scale is represented by a power of 10.
For example, one billion is 1,000,000,000. In scientific notation it is 1 × 109. The exponent 9 tells you that the value is nine powers of ten above 1.
In this Wikimedia Commons visualization, each larger block represents a tenfold increase. It helps you see how repeated multiplication by 10 quickly creates enormous values.
Large values also appear in astronomy. The diameter of Earth is about 12,742 km, which is about 1.2742 × 107 m.
Scientific notation is just as useful for tiny values. A typical atomic scale is around 10-10 m, far smaller than anything you can see directly.
The Bohr-model image is a simplified representation of an atom. It is not drawn to scale, but it provides a useful context for discussing extremely small measurements.
Powers of Ten
A power of ten has the form 10n, where n is an integer called the exponent. Positive exponents represent repeated multiplication by 10, while negative exponents represent repeated division by 10.
100 = 1
101 = 10
102 = 100
103 = 1,000
10-1 = 0.1
10-2 = 0.01
10-3 = 0.001
Notice the pattern: increasing the exponent by 1 multiplies the value by 10. Decreasing the exponent by 1 divides the value by 10.
This graph shows how powers grow for several bases. Focus on the line for base 10 and notice how quickly its values increase.
The Khan Academy video gives a step-by-step introduction to scientific notation and powers of ten.
The Form of Scientific Notation
A nonzero number in normalized scientific notation is written in the form:
a × 10n
Here, a is the coefficient and n is an integer exponent. For a positive number, the coefficient must satisfy 1 ≤ a < 10. For a negative number, the absolute value of the coefficient must satisfy 1 ≤ |a| < 10.
Examples of normalized scientific notation include 6.4 × 105, 2.1 × 10-3, and -7.8 × 102.
The expression 64 × 104 has the correct value, but it is not normalized because the coefficient 64 is not between 1 and 10. Renormalizing gives 6.4 × 105.
The number 0 is normally written simply as 0 because there is no nonzero coefficient that satisfies the normalized form.
Coefficient and Exponent
The coefficient contains the significant digits of the number. The exponent tells you the scale.
In 3.72 × 108, the coefficient is 3.72 and the exponent is 8.
In 4.9 × 10-6, the coefficient is 4.9 and the exponent is -6.
For positive quantities, a larger positive exponent usually means a larger number. A more negative exponent usually means a smaller positive number.
Converting Decimal Numbers to Scientific Notation
To convert a nonzero decimal number into scientific notation, move the decimal point until exactly one nonzero digit is to its left. Then count how many places the decimal point moved.
If the original positive number is 10 or greater, the exponent is positive. If the original positive number is between 0 and 1, the exponent is negative.
Example with a large number:
53,000,000 = 5.3 × 107
The decimal point moved 7 places to the left, so the exponent is 7.
Example with a small number:
0.000042 = 4.2 × 10-5
The decimal point moved 5 places to the right to create the coefficient, so the exponent is -5.
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The place-value chart can help you connect decimal movement with multiplication and division by powers of ten.
This Khan Academy video works through several scientific-notation examples.
Converting Scientific Notation to Decimal Notation
To convert from scientific notation to ordinary decimal notation, use the exponent to determine how the decimal point changes.
A positive exponent means multiply by a positive power of 10, so move the decimal point to the right.
6.25 × 104 = 62,500
A negative exponent means divide by a positive power of 10, so move the decimal point to the left.
8.1 × 10-4 = 0.00081
An exponent of 0 leaves the coefficient unchanged because 100 = 1.
7.4 × 100 = 7.4
A useful check is to estimate the size first. If the exponent is strongly positive, the decimal form should usually be large. If the exponent is negative, a positive coefficient between 1 and 10 will produce a value between 0 and 1.
Comparing Numbers in Scientific Notation
For positive numbers, compare the exponents first. A number with a larger exponent is larger when both coefficients are normalized and positive.
For example:
4.8 × 107 > 9.2 × 106
Even though 9.2 is greater than 4.8, the first number is larger because 107 is ten times 106.
If the exponents are equal, compare the coefficients.
7.1 × 105 > 6.9 × 105
Scientific notation therefore separates two ideas: the coefficient gives detail, while the exponent gives scale.
Calculating with Scientific Notation
Scientific notation is especially useful in calculations because the rules of exponents simplify multiplication and division.
Multiplication
Multiply the coefficients and add the exponents.
(2 × 103) × (4 × 105) = 8 × 108
If the new coefficient is not normalized, adjust it.
(6 × 104) × (3 × 102) = 18 × 106 = 1.8 × 107
Division
Divide the coefficients and subtract the exponents.
(8 × 107) ÷ (2 × 103) = 4 × 104
If the quotient coefficient is smaller than 1 or at least 10, renormalize it.
This Khan Academy example demonstrates multiplication using scientific notation.
Addition and Subtraction
For addition or subtraction, first rewrite the numbers so they use the same power of 10. Only then combine the coefficients.
3.2 × 105 + 4.5 × 105 = 7.7 × 105
If the powers are different, rewrite one number first.
2.0 × 106 + 3.0 × 105 = 2.0 × 106 + 0.3 × 106 = 2.3 × 106
This works because you are combining quantities with the same place-value scale.
Scientific Notation in Measurement
Scientific notation is common in measurements because scientific quantities can span many powers of ten. The International System of Units also uses prefixes such as kilo-, milli-, micro-, and nano- to represent powers of ten.
For example, 1 km = 103 m, 1 mm = 10-3 m, and 1 nm = 10-9 m.
The SI base-units diagram connects scientific notation with real measurements in science and engineering.
The image below shows how powers of ten can describe scales from everyday objects to astronomical distances.
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The scale illustration was created by Pablo Carlos Budassi and is available on Wikimedia Commons under a Creative Commons license.
This Khan Academy word problem uses scientific notation with the speed of light and the distance from the Sun, showing how the notation helps in real calculations.
Calculator and Computer Notation
Many calculators and computers use E notation when there is not enough space to display × 10n. For example:
3.2E5 means 3.2 × 105.
7.1E-4 means 7.1 × 10-4.
Do not treat E as a new mathematical operation. It is a compact display convention for a power of ten. When entering values on a calculator, use the calculator's exponent or EE key if it has one, rather than typing an extra multiplication by 10 unless the device instructions require that.
Common Errors and How to Check Your Work
A common error is leaving a coefficient such as 42 or 0.42 in a final scientific-notation answer. In normalized form, the coefficient's absolute value must be at least 1 and less than 10.
Another common error is choosing the wrong exponent sign. For a small positive decimal such as 0.0007, the exponent must be negative: 7 × 10-4.
A third error is adding exponents when adding numbers. Exponents are added during multiplication, not ordinary addition. For addition and subtraction, first express both numbers with the same power of ten.
You can check an answer by estimating its size. For example, 6.2 × 104 should be tens of thousands, while 6.2 × 10-4 should be much less than 1.
Interactive Tasks
Quiz: Test Your Knowledge
Which expression is in normalized scientific notation? (4.7 times 10 to the sixth power) (!47 times 10 to the fifth power) (!0.47 times 10 to the seventh power) (!4700000 without a power of ten)
What is 42000 in normalized scientific notation? (4.2 times 10 to the fourth power) (!42 times 10 to the third power) (!4.2 times 10 to the third power) (!0.42 times 10 to the fourth power)
What is 0.00056 in normalized scientific notation? (5.6 times 10 to the negative fourth power) (!5.6 times 10 to the fourth power) (!56 times 10 to the negative fourth power) (!0.56 times 10 to the negative fifth power)
What must be true about the coefficient of a positive normalized scientific-notation number? (It is at least 1 and less than 10) (!It is always a whole number) (!It is always greater than 10) (!It is always less than 1)
What does a negative exponent usually indicate for a positive normalized number? (The value is between 0 and 1) (!The value must be negative) (!The coefficient is greater than 10) (!The number has no decimal part)
What decimal number equals 7.3 times 10 to the third power? (7300) (!730) (!0.0073) (!73000)
Which positive number is larger? (4.5 times 10 to the sixth power) (!9.1 times 10 to the fifth power) (!8.8 times 10 to the fourth power) (!7.7 times 10 to the third power)
What is the normalized product of 2 times 10 to the third power and 3 times 10 to the fourth power? (6 times 10 to the seventh power) (!6 times 10 to the twelfth power) (!5 times 10 to the seventh power) (!6 times 10 to the first power)
What is the normalized quotient of 8 times 10 to the sixth power divided by 2 times 10 to the second power? (4 times 10 to the fourth power) (!4 times 10 to the eighth power) (!6 times 10 to the fourth power) (!4 times 10 to the third power)
Why is scientific notation useful? (It makes very large and very small numbers easier to represent and calculate with) (!It changes the actual value of a number) (!It removes the need for place value) (!It can only be used for whole numbers)
Memory Game
| Coefficient | The factor that contains the main digits of a scientific-notation number |
| Exponent | The integer that shows which power of ten is used |
| Power of ten | A value such as ten, one hundred, or one tenth written using base ten |
| Decimal notation | A number written with ordinary place values instead of an exponent form |
| Positive exponent | A scale factor associated with repeated multiplication by ten |
| Negative exponent | A scale factor associated with repeated division by ten |
| E notation | A calculator and computer display form for a power of ten |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Coefficient between one and ten | Normalized scientific notation |
| Positive exponent | Large positive decimal value |
| Negative exponent | Small positive decimal value |
| Add exponents | Multiplication with powers of ten |
| Match exponents first | Addition with scientific notation |
Match each strategy to the type of situation where it belongs.
Crossword Puzzle
| Coefficient | What is the factor containing the main digits in scientific notation? |
| Exponent | What integer tells you the power of ten? |
| Decimal | What kind of point separates whole-number and fractional place values? |
| Magnitude | What word describes the general scale or size of a number? |
| Normalized | What word describes scientific notation with a coefficient whose absolute value is at least one and less than ten? |
| Notation | What word means a system for writing mathematical ideas and numbers? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Scientific notation number hunt: Find eight very large or very small numbers in textbooks, news articles, science websites, or product information, and rewrite each one in normalized scientific notation.
- Powers of ten poster: Create a one-page poster that shows powers from 10 to the negative sixth power through 10 to the sixth power with matching decimal values and one real-world example for at least four powers.
- Scientific notation explanation: Write a short explanation for a classmate showing how to convert 3,600,000 and 0.00036 into scientific notation and explain why the exponent signs differ.
- Scale photo challenge: Take or draw three pictures of objects from very different size scales and label each with a reasonable order-of-magnitude estimate.
Standard
- Measurement interview: Interview a science teacher, technician, engineer, nurse, programmer, or other professional about when very large or very small numbers appear in their work, then summarize how scientific notation could help.
- Calculator E notation investigation: Use two calculators or calculator apps to enter at least six scientific-notation values, record how each display represents the exponent, and explain any differences you notice.
- Scientific notation video tutorial: Produce a two-minute instructional video that teaches one conversion method, includes one large-number example and one small-number example, and ends with a practice question.
- Classroom scale experiment: Measure several classroom objects in meters, convert the measurements to scientific notation, and compare their orders of magnitude.
Advanced
- Astronomy scale project: Research the sizes or distances of four astronomical objects, express all values in the same unit using scientific notation, and create a visual comparison that explains the role of the exponents.
- Microscopic scale project: Research four microscopic structures such as a cell, bacterium, virus, or molecule, express approximate sizes in meters using scientific notation, and evaluate which comparisons are most surprising.
- Data transformation study: Find a public dataset containing values across several powers of ten, convert a sample into normalized scientific notation, and explain how the new representation changes the ease of comparison.
- Science museum field study: Visit a science museum, observatory, laboratory open day, planetarium, or virtual science exhibition, collect examples involving extreme scales, and create a report connecting at least five exhibits to powers of ten.
Learning Assessment
- Conversion reasoning assessment: Convert four decimal numbers into scientific notation and four scientific-notation numbers into decimal notation, then justify the sign and size of every exponent.
- Error analysis assessment: Analyze three incorrect scientific-notation solutions, identify the exact error in each one, correct it, and explain a checking strategy that would prevent the mistake.
- Scale comparison assessment: Order six positive quantities written in mixed decimal and scientific notation from least to greatest and explain how you compared exponents and coefficients.
- Operations assessment: Solve one multiplication, one division, and one addition problem in scientific notation, normalize every result, and explain why the exponent rules differ between the operations.
- Real-world modeling assessment: Choose a scientific or technical quantity, represent it in decimal form and scientific notation, and argue which form communicates the value more effectively.
- Transfer assessment: Interpret three calculator values written in E notation, rewrite them in normalized scientific notation, and explain how you know whether each represents a large or small number.
Evidence of Learning
Knowledge: You can describe the structure of normalized scientific notation, explain positive and negative powers of ten, and connect exponents to place value and scale.
Skills: You can convert between decimal notation and scientific notation, compare quantities, calculate with powers of ten, normalize results, interpret E notation, estimate scale, and check answers for reasonableness.
Products: Useful evidence may include completed practice tasks, posters, annotated images, measurement tables, calculator investigations, videos, interview summaries, research projects, and written error analyses.
Transfer: Strong evidence of learning appears when you use scientific notation correctly in a new context such as astronomy, microscopic measurement, data science, engineering, or another subject and can explain why the notation is useful there.
OERs on the Topic
The English Wikipedia article below gives additional background on scientific notation, conversion, calculator notation, and arithmetic with powers of ten.
Linked Learning Areas
Scientific notation connects place value, exponents, measurement, estimation, algebra, and scientific data. The links below can help you continue learning.