Key takeaway: Scientific notation writes a number as a product of a decimal between 1 and 10 and a power of 10, which keeps very large and very small values short.
Grade 88.EE.A.3Numbers and Operations
Definition of Scientific Notation
Scientific notation repackages a number as a short decimal times a power of 10. It rescues values such as the 93,000,000 miles from Earth to the Sun or the width of a red blood cell, 0.0000072 meter, which are awkward to read and easy to miscount. The size becomes visible in the exponent.
The decimal part must be at least 1 and less than 10. A positive exponent means a large number, found by moving the decimal point right. A negative exponent means a number between 0 and 1, found by moving the point left. The exponent counts how many places the point moved.
Formula and notation
Every number in this form looks like
\[a \times 10^{n}, \qquad 1 \le a < 10, \ n \text{ an integer}\]
So \(93{,}000{,}000 = 9.3 \times 10^{7}\) and \(0.0000072 = 7.2 \times 10^{-6}\).
Key facts
- In scientific notation, positive exponents give numbers of 10 or more; negative exponents give numbers below 1.
- The exponent equals the number of places the decimal point travels.
- To compare two numbers in this form, compare exponents first, then the decimal parts.
- A form such as \(45 \times 10^{5}\) is not finished, since 45 is above 10.
Scientific Notation example
Write the Earth-to-Sun distance, 93,000,000 miles, in scientific notation. Then write the width of a red blood cell, 0.0000072 meter, the same way.
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For 93,000,000: place the point after the first digit to get 9.3. The point moved 7 places left.
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So \(93{,}000{,}000 = 9.3 \times 10^{7}\).
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For 0.0000072: move the point right until 7.2 appears, which takes 6 places.
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Moving right means a number below 1, so \(0.0000072 = 7.2 \times 10^{-6}\).
The distance is \(9.3 \times 10^{7}\) miles and the cell is \(7.2 \times 10^{-6}\) meter wide.
Common mistake
Watch out
Ivy writes 0.00052 as \(5.2 \times 10^{4}\), because the point moved 4 places.
Why it is wrong
She counted correctly but chose the wrong sign. A number smaller than 1 needs a negative exponent: \(0.00052 = 5.2 \times 10^{-4}\). Check by size: \(5.2 \times 10^{4} = 52{,}000\), which is nowhere near 0.00052.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Numbers and Operations