1.6: Scientific Notation
- Page ID
- 127332
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Most of the interesting phenomena our universe has to offer are not on the human scale. It would take about \(1,000,000,000,000,000,000,000\) bacteria to equal the mass of a human body. When the physicist Thomas Young discovered that light was a wave, it was back in the bad old days before scientific notation, and he was obliged to write that the time required for one vibration of the wave was \(1 / 500\) of a millionth of a millionth of a second. Scientific notation is a less awkward way to write very large and very small numbers such as these. Here's a quick review.
Scientific notation means writing a number in terms of a product of something from 1 to 10 and something else that is a power of ten. For instance,
\[\begin{array}{l}
32=3.2 \times 10^1 \\
320=3.2 \times 10^2 \\
3200=3.2 \times 10^3 \ldots
\end{array} \notag\]
Each number is ten times bigger than the previous one.
Since \(10^1\) is ten times smaller than \(10^2\), it makes sense to use the notation \(10^0\) to stand for one, the number that is in turn ten times smaller than \(10^1\). Continuing on, we can write \(10^{-1}\) to stand for 0.1 , the number ten times smaller than \(10^0\). Negative exponents are used for small numbers:
\[\begin{array}{l}
3.2=3.2 \times 10^0 \\
0.32=3.2 \times 10^{-1} \\
0.032=3.2 \times 10^{-2} \ldots
\end{array} \notag\]
A common source of confusion is the notation used on the displays of many calculators. Examples:
\[\begin{array}{ll}
3.2 \times 10^6 & \text { (written notation) } \\
3.2 \mathrm{E}+6 & \text { (notation on some calculators) } \\
3.2^6 & \text { (notation on some other calculators) }
\end{array}\]
The last example is particularly unfortunate, because \(3.2^6\) really stands for the number \(3.2 \times 3.2 \times 3.2 \times 3.2 \times 3.2 \times 3.2=1074\), a totally different number from \(3.2 \times 10^6=3200000\). The calculator notation should never be used in writing. It's just a way for the manufacturer to save money by making a simpler display.
self-check
A student learns that \(10^4\) bacteria, standing in line to register for classes at Paramecium Community College, would form a queue of this size:
The student concludes that \(10^2\) bacteria would form a line of this length:
Why is the student incorrect?