6.3 × 105
The coefficient 6.3 is between 1 and 10, so the expression is in proper scientific notation.
Free Pre-Algebra Guide
Scientific Notation
Learn how powers of ten make very large and very small numbers easier to write, read, convert, compare, and check.
Scientific notation is a compact way to represent numbers by multiplying a coefficient by a power of ten. The format shows both the important digits and the size of the number, which is especially useful for values with many zeros.
Coefficient, Base, and Exponent
A number in scientific notation has the form a × 10n. The coefficient a contains the significant digits. The exponent n tells how many places the decimal moves and in which direction.
4.82 is the coefficient.
Its absolute value is at least 1 but less than 10.
Its absolute value is at least 1 but less than 10.
10 is the base.
Scientific notation always uses a power of ten.
Scientific notation always uses a power of ten.
4 is the exponent.
Move the decimal four places to the right to get standard form.
Move the decimal four places to the right to get standard form.
63 × 104
This value is equivalent, but 63 is too large to be a scientific-notation coefficient.
Powers of Ten
Each time the exponent increases by 1, the value is multiplied by 10. Each time the exponent decreases by 1, the value is divided by 10.
10−4 = 0.0001
10−3 = 0.001
10−2 = 0.01
10−1 = 0.1
100 = 1
101 = 10
102 = 100
103 = 1,000
104 = 10,000
The Zero Exponent Is the Center
Since 100 = 1, positive powers of ten represent values greater than or equal to 10, while negative powers represent positive values less than 1.
Positive and Negative Exponents
When converting from scientific notation to standard form, the exponent sign tells the direction of the decimal movement. The exponent's absolute value tells the number of places.
Move Right
A positive exponent makes the standard-form number larger.
3.4 × 103
= 3,400
Move Left
A negative exponent makes a positive standard-form number smaller than 1.
3.4 × 10−3
= 0.0034
Direction Depends on the Task
The rule above applies when changing scientific notation to standard form. When creating scientific notation, moving the original decimal left gives a positive exponent, while moving it right gives a negative exponent.
Convert Standard Form to Scientific Notation
Move the decimal until exactly one nonzero digit is to its left. Count every place crossed. The direction of the move determines the exponent sign.
48,200.
Original number
4.82
Move left 4 places
1. Place the decimal
Write 4.82 so the coefficient is at least 1 but less than 10.
2. Count the move
The decimal moved four places.
3. Choose the sign
The original decimal moved left, so use a positive exponent.
4. Write the answer
48,200 = 4.82 × 104.
A Small-Number Example
In 0.00071, move the decimal right four places to make 7.1. Moving right gives a negative exponent.
0.00071 = 7.1 × 10−4
Convert Scientific Notation to Standard Form
Begin at the coefficient's existing decimal point. Move right for a positive exponent or left for a negative exponent. Add placeholder zeros whenever the decimal crosses an empty place.
Example: Convert 8.04 × 105
- The exponent is positive, so move right.
- The exponent has an absolute value of 5, so move five places.
- Pass the 0 and 4, then add three placeholder zeros.
8.04 → 80.4 → 804. → 8,040 → 80,400 → 804,000
8.04 × 105 = 804,000
Zeros Hold Empty Places
Do not change or remove the coefficient's digits. Add zeros only when the decimal must move beyond the digits already present.
The Coefficient Rule
In proper scientific notation, the absolute value of the coefficient must satisfy 1 ≤ |a| < 10. That means there is exactly one nonzero digit to the left of the decimal.
| Expression | Proper Form? | Reason or Correction |
|---|---|---|
| 5.2 × 106 | Yes | 5.2 is at least 1 and less than 10. |
| 0.52 × 107 | No | Rewrite as 5.2 × 106. |
| 52 × 105 | No | Rewrite as 5.2 × 106. |
| −7.1 × 10−3 | Yes | |−7.1| is between 1 and 10. |
Equivalent Does Not Always Mean Proper
52 × 105 and 5.2 × 106 have the same value, but only the second expression follows the coefficient rule.
Compare Numbers in Scientific Notation
For positive numbers in proper scientific notation, compare exponents first. A larger exponent means a larger number. If the exponents match, compare the coefficients.
Different exponents
7.1 × 108
is greater than
9.9 × 107
because 8 > 7.
Same exponent
6.4 × 10−5
is greater than
2.8 × 10−5
because 6.4 > 2.8.
Take Care With Negative Numbers
When both values are negative, the number with the greater absolute value is farther left on the number line and is therefore smaller.
Worked Scientific-Notation Examples
Write 6,530,000 in scientific notation
- Move the decimal left six places.
- The coefficient is 6.53.
- Moving left gives a positive exponent.
6.53 × 106
Write 0.0000071 in scientific notation
- Move the decimal right six places.
- The coefficient is 7.1.
- Moving right gives a negative exponent.
7.1 × 10−6
Convert 2.09 × 104
- The exponent is positive.
- Move the decimal right four places.
- Add two placeholder zeros.
20,900
Convert 3.6 × 10−7
- The exponent is negative.
- Move the decimal left seven places.
- Add six zeros before the 3.
0.00000036
Rewrite 45.2 × 103
- Move the coefficient's decimal left once.
- Increase the exponent by 1.
- Check that 4.52 is between 1 and 10.
4.52 × 104
Compare 8.2 × 105 and 3.4 × 106
- Both numbers are positive.
- Compare exponents: 5 < 6.
- The number with exponent 6 is greater.
8.2 × 105 < 3.4 × 106
Common Scientific-Notation Mistakes
Using an Invalid Coefficient
A coefficient such as 32 or 0.32 is not in proper scientific notation. Move its decimal and adjust the exponent until the coefficient's absolute value is at least 1 but less than 10.
Reversing the Exponent Sign
A positive exponent expands a coefficient into a larger standard number. A negative exponent produces a positive number less than 1.
Counting Digits Instead of Places
Count every position the decimal crosses. Do not count the starting position or simply count the visible digits.
Dropping Placeholder Zeros
In 4.2 × 105, the decimal moves five places, so the answer is 420,000—not 42,000.
Multiplying the Coefficient by the Exponent
3 × 104 means 3 × 10,000, not 3 × 4.
Changing the Order of Significant Digits
Decimal movement changes place value, not digit order. The digits in 5.07 must remain 5, 0, 7.
Steps for Solving Scientific-Notation Problems
1. Identify the direction
Decide whether you are creating scientific notation or expanding it.
2. Locate the decimal
A whole number has an understood decimal at its right end.
3. Determine the move
Move until the coefficient is valid, or follow the given exponent.
4. Count every place
The exponent's absolute value must equal the number of places moved.
5. Use the correct sign
Connect the sign to both the task direction and the size of the result.
6. Check the result
Convert back and confirm the digits, decimal location, and overall size.
Final check: the coefficient must satisfy
1 ≤ |a| < 10, and the exponent must match both the
distance and direction of the decimal move.
Practice Scientific Notation
Work through both conversion directions, choose the decimal movement, count places, and check your final answer with guided feedback and hints.
Open the Scientific Notation ToolRelated Math Resources
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