সূচকীয়করণ এবং বিশালতার মাপ

The order of magnitude is the power of $10$ a number is raised to when it's in scientific notation.

For example, the order of magnitude for $19{,}400$ would be $4$. This is because when we convert it into scientific notation, $1.94\times10^4$, the $10$ is being raised to the $4^{\text{th}}$ power.

We can also compare orders for two different numbers in scientific notation. Consider for example

$$A = 6.7 \times 10^8$$ $$B = 6.7 \times 10^9$$ $$C = 6.7 \times 10^{10}$$

One can quickly see that $B$ is a bigger number than $A$, and $C$ is bigger than $B$. However, we can say more. We can say $C$ is two orders of magnitude larger than $A$. Also, $C$ is ten times the smaller number $B$; $B$ is ten times $A$.

Now it's your turn to try a problem yourself.

To find the order of magnitude, we need to put this in scientific notation first. $$6{,}032{,}095=6.03209\,5\times10^6$$ Since $10$ is raised to the $6,$ the order of magnitude is $6$. We get to ignore the ugly decimal before the $10^6$.

It's important that the number is in scientific notation. Currently it is not because $63.2$ is not between $1$ and $10$. We have to move the decimal place to the left once more. This will also raise the power of $10$ by $1$. $$6.32\times10^9$$ Therefore, the order of magnitude is $9$.

(Video) Orders of Magnitude

by Truckee

Truckee has a good video about orders of magnitude.

$\large{\text{Summary}}$

We use orders of magnitude for estimating. If you're looking for the volume of a lake or the area of a field, you can use estimates. Sometimes the numbers will be written in scientific notation. Other times you'll practice writing a number in scientific notation yourself. Regardless, these estimates are useful for small numbers and large numbers alike. The power of $10$ could be positive or negative.

If you're asked to find the order of magnitude for $2.3\times10^4$ you have two options. The first is rounding the scientific notation to the ones place. $$2.3\times10^4\approx 2\times10^4$$ The other, is to just write out the power of ten. $$2.3\times10^4\approx10^4$$

The numbers $2 \times 10^4$ and $10^4$ are certainly different. The first number is $2$ times larger than the second. However, the powers of $10$ are the same. They are numbers of the same order. In addition, there are a total of four factors of ten.

She then looks at an example.

Estimate the volume of water in Gilmore Lake in the Desolation Wilderness. The lake is $2 \text{ km}$ across and has an average depth of $50 \text{m}$.

We can think of the lake as a cylinder with a height of $50 \text{ m}$ and a radius of $1\text{ km}$. The volume of a cylinder is: $$V=\pi r^2h$$ We can write $1 \text{ km}$ as $1\times10^3$ and $50 \text{ m}$ as $5\times10^1$. Plugging these into the equation, we get: $$\begin{align} V&=\pi r^2h\\ V&=\pi(10^3)^2(5\times10^1)\\ V&=\pi\times10^6\times5\times10^1\\ V&=5\pi\times10^{6+1}\\ V&=5\pi\times10^7 \end{align}$$ Be careful though. We see that $5\pi$ is approximately $15.7$. This is not between $1$ and $10$. In fact, $5\pi$ is greater than $10$.

$$V\approx1.57\times10^8$$

We can estimate this more as $2\times10^8$ or just $10^8$. These are called examples of approximate comparisons.