Algebra Chapter 1 Section 7: Scientific Notation
In science, there are many numbers which are either exceedingly large (for example, the number of water molecules in a liter of water) or exceedingly small (for example, the charge, in Coulombs, of a single electron). This is the nature of the reality we live in. This is why, in some cases, it is easier to read and right these extremely small and large values in a more compact notation. Yes, this does somewhat reduce accuracy, but the loss of accuracy has very little baring on the accuracy of the final answer. The most common compact notation in Social Sciences, Life Sciences, and Physical Sciences is called Scientific Notation.
The notation for Scientific Notation is $N \times 10^{n}$, where N is a number which is at least 1 but less than 10 $[1 \le N < 10]$ and n is the magnitude of the number. There are a couple steps to convert to scientific notation:
As a first example, there is a number in Chemistry called the mole. It is defined as the number of Carbon-12 in exactly 12 grams of ultra-pure carbon-12, and the number associated with the mole is approximately 602200000000000000000000. The first step of reducing this to Scientific Notation is to put a decimal point in between the first and the second digit of the number so that it becomes 6.02200000000000000000000. The second step is to multiply it by $10^{n}$, where n is the number of digits which the decimal point was moved. Since the initial number is greater than 1, n is positive. In this case, there were 23 digits after the 6 and before the end of 602200000000000000000000, so n=23. That makes the number $6.02200000000000000000000 \times 10^{23}$. The third step is to round the number as desired, so now it becomes $6.022 \times 10^{23}$.
Another good example found in Physics is the Elementary Coulomb Charge, the smallest charge a sub-atomic particle can have without getting to the quark level. The value is 0.000000000000000000160C. This can also be converted to Scientific Notation with the same steps as above. Step 1 is still putting the decimal point between the first two significant digits. 1.60. The second step is still to multiply it by $10^{n}$, where n is the number of digits which the decimal point was moved. In this case, n is going to be 19, but since the number is less than 1, it will be negative since it is equivalent to dividing the number by $10^{19}$. If you remember from last time, $\frac{1}{x^{n}}=x^{-n}$. So the fundamental charge, written in scientific notation, is $1.60 \times 10^{-19} C$.
I've been mentioning significant digits a lot in this lecture series, and it will also come up frequently in Physics and Chemistry (and all of science). A significant digit is a digit which is significant to determining the value of a number. In order to determine the significant digits, first convert to scientific notation. All non-zero digits are significant. All zeroes between non-zero numbers are significant. All leading zeroes are NOT significant. Any trailing zero of the non-approximation are significant, but any zero of non-approximations are insignificant. For example, for the quantity for the mole, since $6.02200000000000000000000 \times 10^{23}$ is an approximation, all zeroes after the second 2 are insignificant, since the number is approximated. If I rounded to $6.0 \times 10^{23}$ instead of $6.022 \times 10^{23}$, then the zero would be significant since it does say something about the accuracy of the number.
When rounding after a calculation, it would be a good idea to know the precision of the numbers your working with. For physical and life sciences, this depends on the precision of the instruments you're using. For social sciences, this depends on other parameters. For both cases, it's a matter of not using anything more precise than the precision of the least precise value. The rules for this are a little different for addition/subtraction than they are for multiplication/division. The reason for this is because of the magnitude changes for the two types of operation. For the first, it's hard to have a large change, but for the latter, it's hard to have a small change.
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The notation for Scientific Notation is $N \times 10^{n}$, where N is a number which is at least 1 but less than 10 $[1 \le N < 10]$ and n is the magnitude of the number. There are a couple steps to convert to scientific notation:
- Place a decimal point in between the first two significant digits of the number.
- Find the magnitude by determining how many digits which the decimal point has moved in step 1.
- If the initial number is greater than 1, n is positive.
- It the initial number is less than 1, n is negative.
- Round N to the appropriate level
As a first example, there is a number in Chemistry called the mole. It is defined as the number of Carbon-12 in exactly 12 grams of ultra-pure carbon-12, and the number associated with the mole is approximately 602200000000000000000000. The first step of reducing this to Scientific Notation is to put a decimal point in between the first and the second digit of the number so that it becomes 6.02200000000000000000000. The second step is to multiply it by $10^{n}$, where n is the number of digits which the decimal point was moved. Since the initial number is greater than 1, n is positive. In this case, there were 23 digits after the 6 and before the end of 602200000000000000000000, so n=23. That makes the number $6.02200000000000000000000 \times 10^{23}$. The third step is to round the number as desired, so now it becomes $6.022 \times 10^{23}$.
Another good example found in Physics is the Elementary Coulomb Charge, the smallest charge a sub-atomic particle can have without getting to the quark level. The value is 0.000000000000000000160C. This can also be converted to Scientific Notation with the same steps as above. Step 1 is still putting the decimal point between the first two significant digits. 1.60. The second step is still to multiply it by $10^{n}$, where n is the number of digits which the decimal point was moved. In this case, n is going to be 19, but since the number is less than 1, it will be negative since it is equivalent to dividing the number by $10^{19}$. If you remember from last time, $\frac{1}{x^{n}}=x^{-n}$. So the fundamental charge, written in scientific notation, is $1.60 \times 10^{-19} C$.
I've been mentioning significant digits a lot in this lecture series, and it will also come up frequently in Physics and Chemistry (and all of science). A significant digit is a digit which is significant to determining the value of a number. In order to determine the significant digits, first convert to scientific notation. All non-zero digits are significant. All zeroes between non-zero numbers are significant. All leading zeroes are NOT significant. Any trailing zero of the non-approximation are significant, but any zero of non-approximations are insignificant. For example, for the quantity for the mole, since $6.02200000000000000000000 \times 10^{23}$ is an approximation, all zeroes after the second 2 are insignificant, since the number is approximated. If I rounded to $6.0 \times 10^{23}$ instead of $6.022 \times 10^{23}$, then the zero would be significant since it does say something about the accuracy of the number.
When rounding after a calculation, it would be a good idea to know the precision of the numbers your working with. For physical and life sciences, this depends on the precision of the instruments you're using. For social sciences, this depends on other parameters. For both cases, it's a matter of not using anything more precise than the precision of the least precise value. The rules for this are a little different for addition/subtraction than they are for multiplication/division. The reason for this is because of the magnitude changes for the two types of operation. For the first, it's hard to have a large change, but for the latter, it's hard to have a small change.
- When adding or subtracting two numbers, the rule for rounding is to round to the same decimal place as the number with the least amount of digits to the right of the decimal place. For example, if we have 115.2+4.889, the final answer will be 121.089, but since 115.2 has the fewest digits to the right of the decimal point, the final answer should be rounded to have that few digits to the right of the decimal point, 121.1.
- When multiplying or dividing, the rounding is determined from the number with the fewest significant digits total. For example, if we multiply 15.88 with 3.14, then the answer is 49.86, but since 3.14 has fewer significant digits, the final answer has to have 3 significant digits, so the answer is 49.9.
That's the end of this section. If you have any questions, please leave them is the comments. Like and share this post if you found it helpful. And until next time, stay curious.
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