什么是科学计数法？

(Video) Scientific Notation

by mathman1024

This video by mathman1024 explains scientific notation.

$\large{\text{Summary}}$

Scientific notation is a way to express really large or really small numbers. These numbers include multiplying ten raised to some power. No matter what power we use, the number we write will be equal to the original number. We're not changing the value of anything.

Scientific notation looks like this: $$M\times10^n\\ \text{where }1\leq M<10$$

Let's look at an example of this. Here's a number in scientific notation: $$\boxed{1.7\times10^{-15}}$$

Here's the number in decimal notation. $$1.7\times10^{-15}\\ =\boxed{0.00000\,00000\,00001\,7}$$ The exponent on the $10$ tells you how many places you move the decimal point (and how many powers of ten there are). In this example we moved the decimal place to the left $15$ spaces. The significant figures are the digits that appear starting with the first nonzero digit, moving left to right. For this number, the significant digits are the $1$ and the $7$.

The video shows a neat example to illustrate how small a number this is. Suppose every second of your life you are given $\$1.7 \times 10^{-15}$. How much money will you have at the end of your life from this? For sake of argument, let's suppose you live for $100$ more years.

$$\begin{array}{c|c|c} 100\text{ years}&365\text{ days}&24\text{ hours}&3{,}600\text{ seconds}\\ \hline &1\text{ year}&1\text{ day}&1\text{ hour} \end{array}$$ $$\begin{array}{c|c|c} 100\color{#29A603}{\cancel{\color{black}{\text{ years}}}}&365\color{#29A603}{\cancel{\color{black}{\text{ days}}}}&24\color{#29A603}{\cancel{\color{black}{\text{ hours}}}}&3{,}600\text{ seconds}\\ \hline &1\color{#29A603}{\cancel{\color{black}{\text{ year}}}}&1\color{#29A603}{\cancel{\color{black}{\text{ day}}}}&1\color{#29A603}{\cancel{\color{black}{\text{ hour}}}} \end{array}$$

So there are $100\cdot365\cdot24\cdot3{,}600$ seconds per $100$ years, or $3{,}153{,}600{,}000$ seconds, or $3.1536\times10^9$ seconds. And if you got $\$ 1.7\times10^{-15}$ each of those seconds, then at the end of $100$ years you would have $$\begin{align} 3{,}153{,}600{,}000&\\ \times\ 0.00000\,00000\,00001\,7&\\ \hline 0.00000\,5& \end{align}$$ You wouldn't even make one cent. That's how small that number is.

Let's see an example of a humongous number now: $2.52\times10^8$. Since the exponent is positive we move the decimal place to the right by $8$. $$2.52\times10^8\\ =252{,}000{,}000$$

So a negative power is a tiny number. A positive power can be a large number.

Let's now try converting a number into scientific notation. The example he gives is, $0.00000\,00000\,809$.

First, remember we want to create a decimal between $1$ and $10$. So where should the decimal point move to? Between the $8$ and the $0$ is the correct answer. This would give us the decimal number $8.09$ which is in between $1$ and $10$. If you move it over in one decimal point in either direction you can see that the resulting number isn't between $1$ and $10$.

To figure out the exponent, just count the number of spaces the decimal has to move in order to get between the $8$ and $0$ that we identified above.

Remember that the exponent tells you which direction to move the decimal. So $8.09\times10^{11}$ tells us that the decimal should be moved to the right $11$ spaces in order to get our original number. But our original number is $11$ spaces to the left. So in fact the correct answer is: $$8.09\times10^{-11}$$

Try converting this last one on your own and then watch the video to see the answer.

1. $9{,}200{,}000{,}000{,}000$

(Video) Scientific Notation

by Veritasium

Veritasium is a great channel about science. Here is a good explanation of what scientific notation is, when we use it, and why we need it.

$\large{\text{Summary}}$

We use scientific notation (or scientific form) to represent really big numbers. One example they give in the video is the mass of the sun. $$2{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000\text{ kg}$$ This is $2$ followed by $30$ zeros.

This notation takes advantage of the powers of $10$. $$10^2=10\times10=100\\ 10^3=10\times10\times10=1{,}000$$ Notice that $10^2$ has $2$ zeros. Similarly, $10^3$ has $3$ zeros.

So if the mass of the sun has $30$ zeros, we can use $10^{30}$. But don't forget the $2$. We can multiply this on in front. $$2\times10^{30}\text{ kg}$$

We sometimes have to deal with really small numbers. The example they give is the mass of a proton. $$0.00000\,00000\,00000\,00000\,00000\,01673\text{ kg}$$ That's a zero point and $26$ zeros, followed by $1{,}673$.

We can use scientific form for this as well. $$10^{-1}=\frac1{10}=0.1\\ 10^{-2}=\frac1{100}=0.01\\ 10^{-3}=\frac1{1{,}000}=0.001$$ The exponent tells you how many places to the right the 1 is.

So if the mass of a proton is $26$ zeros followed by a $1{,}673$ we can write this as: $$1.673\times10^{-27}$$ We add one more to $26$ because we want it to be the place where the $1$ is, which will be one more over. When it's easier to do so, we can also multiply or divide two numbers that are in scientific notation.

Let's think about another example of a really large number. Look at the picture above of the world with faces on it. Now, obviously, there are too many people in the world to put all their faces on the globe, but we can imagine that all of people in the world are represented. The world population is around $7$ billion $(7{,}000{,}000{,}000)$ people.

Scientific Notation: Moving the Decimal

The easiest way to remember how scientific notation works is illustrated below.

If there is a $\color{red}{\text{negative}}$ exponent, you move the decimal to the left because negative numbers are to the left on the number line.

If there is a $\color{blue}{\text{positive}}$ exponent, you move the decimal to the right because positive numbers are to the right on the number line.

In the first example, there are $5$ digits to the left of the decimal point. The second example has no decimal point because our final answer is a really big whole number.