Scientific Notation
Scientific notation is a way of writing very large or very small numbers as a number between 1 and 10 multiplied by a power of 10. In brief: a number is in scientific notation when it has the form a × 10n with 1 ≤ a < 10 and n an integer. For example, 4,500,000 = 4.5 × 106 and 0.00032 = 3.2 × 10−4.
The notation keeps numbers short, makes their size easy to compare and avoids counting zeros. It is used in science, engineering and on calculators, which show results such as 3.2E−4. This guide explains how to convert in both directions, how to calculate with scientific notation and how to avoid the usual errors.
What is scientific notation?
In the form a × 10n, the number a is called the coefficient or significand, and n is the exponent. The coefficient must be at least 1 and less than 10, so there is exactly one nonzero digit before the decimal point. Then the exponent n tells the size, or order of magnitude, of the number.
A positive exponent means a large number, and a negative exponent means a number between 0 and 1. The expression 0.5 × 103 is not in standard scientific notation, because 0.5 is less than 1. The correct form of 500 is 5 × 102.
How do you convert a number to scientific notation?
- Place the decimal point after the first nonzero digit.
- Count how many places the decimal point moved.
- Moving left gives a positive exponent. Moving right gives a negative exponent.
- Write the coefficient followed by × 10 to that exponent.
Example 1: a large number
Convert 73,200,000. The new decimal point goes after the 7, giving 7.32. It moved 7 places to the left, so the exponent is 7: 73,200,000 = 7.32 × 107.
Example 2: a small number
Convert 0.000406. Place the point after the 4, giving 4.06. It moved 4 places to the right, so the exponent is −4: 0.000406 = 4.06 × 10−4.
How do you convert back to a normal number?
Move the decimal point by the number of places given by the exponent: right for positive exponents and left for negative exponents. Add zeros as needed. For example, 2.5 × 105 = 250,000 and 9 × 10−3 = 0.009.
How do you calculate with scientific notation?
Multiplication: multiply the coefficients and add the exponents. (3 × 104)(2 × 105) = 6 × 109. If the coefficient becomes 10 or more, rewrite it: (4 × 103)(5 × 102) = 20 × 105 = 2 × 106.
Division: divide the coefficients and subtract the exponents. (8 × 107) ÷ (2 × 103) = 4 × 104.
Addition and subtraction: first write both numbers with the same exponent. For 3.2 × 104 + 5 × 103, rewrite the second as 0.5 × 104. The sum is 3.7 × 104.
Scientific notation examples table
| Ordinary number | Scientific notation | Exponent |
|---|---|---|
| 0.0000001 | 1 × 10−7 | −7 |
| 0.00052 | 5.2 × 10−4 | −4 |
| 0.8 | 8 × 10−1 | −1 |
| 7 | 7 × 100 | 0 |
| 340 | 3.4 × 102 | 2 |
| 91,000 | 9.1 × 104 | 4 |
| 6,020,000 | 6.02 × 106 | 6 |
| 15,000,000,000 | 1.5 × 1010 | 10 |
Where does scientific notation come from?
Writing large numbers as a product with powers of ten has roots in ancient attempts to name huge quantities. Archimedes, in his work on counting grains of sand, built a system for very large numbers using powers of the myriad (10,000). The modern form depends on exponent notation, which became standard in the seventeenth century, and on the decimal system. The term “scientific notation” and its widespread use grew with science and engineering and later with calculators and computers.
Where is scientific notation used today?
Astronomy uses it for distances: the Sun is roughly 1.5 × 108 km from Earth, which is an approximate average. Chemistry uses it for Avogadro’s number, about 6.022 × 1023 particles per mole. Biology and engineering use it for tiny sizes: a typical bacterial cell is on the order of 10−6 m. Calculators and programming languages print it as 1.5E8 or 1.5e8.
Which mistakes are common?
Coefficient outside the range
12 × 103 and 0.4 × 105 are not in standard form. Adjust them to 1.2 × 104 and 4 × 104.
Wrong sign of the exponent
Numbers smaller than 1 have negative exponents. 0.03 is 3 × 10−2, not 3 × 102.
Adding without matching exponents
3 × 105 + 2 × 103 is not 5 × 108 or 5 × 105. Convert to the same exponent first: 300,000 + 2,000 = 302,000 = 3.02 × 105.
Counting places from the wrong side
Count from the new decimal position to the original one. Checking the answer by estimating its size avoids most errors.