How to find the number of significant figures?

**What is significant figures?**

The number of significant figures arising from this is essentially the number of figures considered to have a degree of reliability.

The simplest method for calculating significant digits is by first determining whether or not a number has a decimal point.This rule is referred to as the *Atlantic-Pacific Rule**.*

**What are the Rules for Determining Number of Significant Figures?**

1) All nonzero digits are significant.

2) With two special cases zeros are significant as well:

- Zeros before the decimal point.
- Zeros after the decimal point and before the first nonzero digit.

3) An ambiguous case is the Terminal zeros before the decimal point in amounts greater than one.

**Rules for numbers without a decimal point**

1. Start counting for sig. figs. A non-zero digit on the first.

2. STOP counting for sig. figs. On the non-zero last digit.

3. Non-zero digits are STILL significant

4. Zeroes in between two non-zero digits are significant. All the remaining zeroes are insignificant.

**Rules for Numbers WITH a Decimal Point**

1. Start counting for sig. figs. A non-zero digit on the first.

2. STOP counting for sig. figs. On the very last digit (regardless last digit is a zero or non-zero).

3. Non-zero digits are always significant.

4. Any zero is still significant since the first non-zero digit. The zeroes preceding the first non-zero digit are insignificant.

**1) Integers: **

The zeros toward the end of an integer do not count as significant.

If you have $1000 in your wallet, it could vary from over $500 to under $1500. The precision of 1000 is to one sig. fig.

If you have $1100 in your wallet, it could vary from over $1050 to under $1150. The precision of 1100 is to two sig. fig.

If you have $1110 in your wallet, it could vary from over $1105 to under $1115. The precision of 1110 is to three sig. fig.

If you have $1111 in your wallet, it could vary from over $1110**.**5 to under $1111**.**5. The precision of 1111 is to four sig. fig.

Note: As you may have noticed, the last zeros of an integer number do not count as significant.

Example:

1) 107: It has 3 significant figures.

2) 10,700: It has 3 significant figures.

3) 1,040,700: It has 5 significant figures.

**2) Decimals**

All figures are significant except the leading zeros after its decimal point for a decimal number that is less than 1. For instance 0.623 kg is less than 1 and has 3 sig. figs. This is identical to 623 grams.

Suppose we have 0.000623 kg instead. The three zeros before 623 but after the decimal point are not considered to be significant. This is equal to 623 milligrams and has three sigs. figs.

Now, if the decimal number is greater than 1, for example, 6,789 kg, it has 4 sig. figs. This is identical to 6,789 grams and has four sigs. figs.

Assume now we have 8.0235kg now, which can be written as 8.023.5 grams. It has got 5 sig. figs.

Here, the zero after its decimal point, however leading to the 235 part, considers as significant in light of the fact that the number is greater than 1 and has a non-decimal part that is 8.

#### How To Find The Number of Significant Figures? With Example

1**)** 0**.**13070kg: It has 5 significant figure.

2**)** 1**.**07000cm: It has 6 significant figure.

3**)** 0**.**0007cm: It has 1 significant figure.

4**)** 22**.**0000cm: It has 6 significant figure.

5**) **0**.**000009cm: It has 1 significant figure.

6**)** 1**.**0400700cm: It has 8 significant figure.

7) 10**.**407005cm: It has 8 significant figure.

8) 100**.**000, 0020cm: It has 10 significant figure.

9) 100**,**000,000**.**0cm: It has 10 significant figure.

10) 100**,**000,000**,**000cm: It has 1 significant figure.

**I) For multiplication, division,** raising to a power, or taking any roots,

a) The final result must be rounded to the same number of sig. figs. if the given numbers have the same number of sig. figs.,

b) The final result must be rounded to the lowest number of sig. figs. if the given numbers have different numbers of sig. figs.,

For instance, if 12 (2 sig. figs.) is duplicated by 25 (likewise 2 sig. figs.), the subsequent number is 300 however should be written in a form that shows it is good to 2 sig. figs. You may either put a small bar on the zero after 3, or compose the number as 3.0×102.

As another example, if 12**.**0 (3 sig. figs.) is multiplied by 25 (2 sig. figs.), the total must be written in two significant figure again as 300 with a tiny bar on the zero after 3 or as 3**.**0x10^{2}.

**II) The precision of the numbers added or subtracted is significant for addition or subtraction.**

Example, if the mass of the screw is measured with a scale that is good to one gram of precision, and the mass of the corresponding nut is measured with another scale that is good (or precise) to one milligram (1000 times better precision), and if we want to add the mass of the two, the high precision of the nut mass is useless compared to the low precision of the bolt mass!

Let,

Mass of the bolt= 8 grams (8000milligrams, 1 sig. fig.)

Mass of the nut= 0**.**675 grams (675milligrams 3 sig. fig.)

Total= 8.675 gms

However, the total may not be written as 8**.**675grams! First, we should round off 0**.**675milligrams and that will be 1 gm. After that addition is done and total mass will be 9 gms i.e. (9000mg). Thus, it end up with 1 sig fig.

**Examples:**

1) 75m x 4m = 3×10^{2}m^{2} (1 sig. fig.)

2) 75cm x 4.0cm =3**.**0x10^{2}m^{2} (2 sig. fig.)

3) 0**.**750 ft x 4.000 ft =3**.**00ft^{2} (3 sig. figs.)

4) 7500 in. x 0.004 in. = 3**.**0x10^{1} in^{2} (2 sig. fig.)

5) 125m / 25s = 5.0m/s (2 sig. fig.)

6) 80f t / 16s =5ft/s (1 sig. fig.)

7) 33,333mi / 3h =10,000 mi/h (1 sig. fig.),

8) 3750km / 2.50s =1**.**50×10^{3} km/s (3 sig. figs.)

9) (25m – 16m) / 0.0003s = 30000 m/s (1 sig. fig.)

**Another Rule to determine the significant figure: The Pacific Rule & the Atlantic Rule**

It tends to be difficult to recall all the rules regulating significant figures and if each zero is significant or not significant. Another way to assess significant figures (sig figs) is the Pacific and Atlantic Rule. This is one of the easiest method to determine significant digits.

Use the Pacific rule (note the double P’s) whenever a number has a decimal present. The Pacific Ocean is centered on the left side of the United States, so start with the number on the left side. At the first non-zero number, begin counting Sig Figs and continue to the end of the number. For instance, since 0.000560 has a decimal number, begin from the left side of the number. After the first non-zero number (5), try not to start counting Sig Figs, then count all the way to the end of the number. There are 3 sig figs in this number, thus (5, 6, and 0).

Use the Atlantic rule (again, remember double A’s) if a number does not have a decimal (the decimal is absent). Since the Atlantic Ocean is on the right side of the United States, start at the first non-zero number on the right side of the number and begin counting sig figs. For instance, since there is no decimal in 2900, start from the right side of the number and begin counting sig figures at the first non-zero number. So in this number, there are two Sig Figs (2, 9).

In short, the rule specifies that if there is no decimal point, then the Atlantic/right side zeroes are insignificant. If a decimal point is present, so the Pacific/left side zeroes are insignificant.