Scientific notation is a compact way to write very large and very small numbers using powers of ten. This cheat sheet helps students convert between standard form and scientific notation, compare values, and perform operations. It is useful for math, science, measurement, astronomy, and any topic where numbers can be extremely large or tiny.
The main idea is to write a number as , where and is an integer. Multiplying and dividing numbers in scientific notation uses exponent rules for powers of ten. Adding and subtracting usually requires matching the powers of ten before combining the decimal parts.
Key Facts
- A number in scientific notation has the form , where and is an integer.
- Moving the decimal left gives a positive exponent, such as .
- Moving the decimal right gives a negative exponent, such as .
- To multiply, multiply the decimal factors and add exponents: .
- To divide, divide the decimal factors and subtract exponents: .
- To add or subtract, rewrite numbers with the same power of ten before combining the decimal factors.
- After any operation, rewrite the answer so the first factor is at least and less than .
- For powers of ten, , , and .
Vocabulary
- Scientific notation
- A way to write a number as , where and is an integer.
- Coefficient
- The decimal factor in scientific notation, such as in .
- Power of ten
- An expression like that shows repeated multiplication or division by .
- Exponent
- The number in that tells how many places the decimal point moves.
- Standard form
- The ordinary way to write a number without powers of ten, such as or .
- Order of magnitude
- A comparison based on powers of ten, where each increase of in the exponent means the number is times larger.
Common Mistakes to Avoid
- Using a coefficient greater than or equal to , such as , is wrong because scientific notation requires .
- Giving a negative exponent for a large number is wrong because numbers greater than usually need a positive exponent in scientific notation.
- Adding exponents when adding numbers, such as treating as , is wrong because exponent rules for adding do not work that way.
- Forgetting to renormalize after multiplication can leave an answer like , which should be written as .
- Subtracting exponents in the wrong order during division changes the size of the answer, so uses , not .
Practice Questions
- 1 Write in scientific notation.
- 2 Compute and write the answer in scientific notation.
- 3 Compute and write the answer in scientific notation.
- 4 Explain why is not written correctly in scientific notation, and describe how to fix it.
Understanding Scientific Notation & Operations Reference
Scientific notation works because our base ten number system is built from place value. Each move one place to the left makes a digit worth ten times more. Each move one place to the right makes it worth one tenth as much.
The exponent records how far the decimal point has been shifted, so it describes the scale of the number. The first number records the important digits.
This separation is useful because it prevents long strings of zeros from hiding the information that matters. For example, measurements of one point zero two times ten to the fourth and one point zero two times ten to the seventh have the same leading digits, but their sizes are very different.
When multiplying or dividing, think about the job done by each part of the number. The decimal factors determine the ordinary multiplication or division. The powers of ten determine how the size changes.
A useful check is to estimate the answer before calculating. If two quantities are each around ten thousand, their product should be around one hundred million. If your written answer is close to one tenth or ten times that estimate, an exponent rule was probably used in the wrong direction.
Sometimes a calculation produces a value such as seventy two times ten to the fifth power. It has the correct value, but moving the decimal one place left requires increasing the exponent by one. This keeps the value unchanged while putting it in standard scientific notation.
Addition and subtraction need more care because the numbers must refer to the same sized place-value groups. This is similar to adding three meters and twenty centimeters. You first express both amounts in one unit.
Suppose a problem combines three point four times ten to the sixth with two point five times ten to the fifth. The second quantity is two tenths as large in scale, so it can be written as zero point two five times ten to the sixth. Now the decimal factors represent the same power of ten and can be combined.
Students often make the mistake of adding exponents during addition. That rule belongs to multiplication, not addition. Writing each number in expanded words can help reveal whether an answer makes sense.
Scientific notation appears whenever scientists handle values outside everyday sizes. A biologist may describe the mass of a cell. A physicist may work with the wavelength of light.
Astronomers measure distances that make ordinary notation hard to read. It is important to notice significant digits in these settings. A measurement written as six point zero times ten to the third power usually communicates more precision than six times ten to the third power.
On calculators, use the exponent key carefully, often marked with letters such as E or EE. A display like four point two E negative six means four point two times ten to the negative sixth power. Read the negative exponent as a small quantity, not a negative quantity.