Significant Figures/Digits & Rules

Significant Figures

Tell you how well a number is known, how precise it is. This is called precision.

Precision

The precision or uncertainty of the number is determined by the last significant digit of the number. (In science, this is very important because science relies on measurements and measurements always have some uncertainty to them).

16.5

The last significant digit is 5, the precision (or uncertainty) of the number is in the tenths place. This means that the value of the number lies between 16.4 and 16.6.

All Non Zero Digits

Are significant. (1234).

All Internal/Captive Zeroes Between Two Non-Zero Digits

Are significant. (1230450067).

Leading Zeroes, The Zeroes Infront of the First Non Zero Digit

Are NOT significant. (0000.004567, 007865).

Trailing Zeroes, The Zeroes After the Last Non-Zero Digit

Are ONLY significant when the number has a DECIMAL point AND are after the last non zero digit. (2.00, 000.34500, 0.000345000000).

How Many Sig Figs in 89400?

3 Sig Figs. 89400 (only 8, 9 and 4 are significant as they are non-zero digits. The trailing zeroes are not significant as there is no decimal point).

How Many Sig Figs in 230.00?

5 Sig Figs. ( 230.00 In this case, all the digits, including the trailing zeroes are significant as there is a decimal point and they are after the last non-zero digit).

Exact Numbers

Have unlimited significant figures. This applies to numbers that can be counted (like eggs in a carton), some numbers in equations (2πr, the 2 is exact), and numbers by definition (1 inch. is EXACTLY 2.54 cm). Note that the WORDING in a problem can sometimes tell you if a number is exact or not. Look for words like EXACTLY (exact numbers), ABOUT and APPROXIMATELY (means that it is an estimate, therefore there is a numbered amount of sig figs, not unlimited).

There are 48 students in a class room. How many Sig Figs?

Unlimited, exact. Since this number can be counted and is exact, it has unlimited figures, not just two.

There are 345 pages in a book. How many Sig Figs?

Unlimited, exact. This is an exact number that can be counted as well.

The number 2 in the formula 2πr. How many Sig Figs?

Unlimited, exact. The number 2 in this formula is exact.

There are 1,000 meters in a kilometer. How many Sig Figs?

Unlimited, exact. This is because it is an exact conversion by definition.

There Are About 14 miles between NYC and Montclair. How many Sig Figs?

2 Sig Figs. The word ABOUT tells you that the measured distance is an estimate, therefore it is not unlimited.

Irrational Numbers

Have the same number of significant figures as are shown in the number. EX: π is the most commonly used irrational number in science. Any given value for π is an APPROXIMATION, therefore, its significant figures are determined by how the number is approximated.

How many Sig Figs in π = 3.14?

3, because π is an irrational number (a number that is approximated) and is based on how many numbers are shown in the problem.

How many Sig Figs in π = 3.14159?

6 Sig Figs. π is an irrational number (a number that is approximated) and is based on how many numbers are shown in the problem.

For Numbers Written in Scientific Notation

The significant figures are determined by the coefficient (mantissa), the number in front of the power of 10.

How Many Sig Figs in 1.456 × 10^6?

4 Sig Figs. This is because you only need to look at the coefficient of the problem, not the power of 10. 1.456 has 4 significant figures.

How Many Sig Figs in 130.?

3 Sig Figs. Remember, IF there is a DECIMAL and the zero is AFTER a non-zero digit, that zero is a SIG FIG.

How Many Sig Figs in 7500 × 10³?

2 Sig Figs, the zeroes after the last non-zero digit is not significant as there is no decimal.

The 653 of a book with 653 pages?

Exact, Unlimited. The pages can be counted.

When Two Different Unit Systems are Being Used to Convert

The answer is usually NOT exact (with some exceptions, some conversions are EXACT by definition). When you convert between two systems, like meters to miles, the answer is an APPROXIMATION.

Conversions WITHIN their own systems

Are exact, like cups to gallons.

The 1609 in 1 mile is 1609 meter?

4 Sig Figs. This is an approximation and a conversion between two unit systems.

The 366 days in a Leap Year

Exact, Unlimited Sig Figs as this number can be counted on a calendar.

How Many Sig Figs in 0.001000?

4 Sig Figs.

Rules for Sig Figs AFTER Multi/Divid

The number of significant figures in your answer is determined by the term with the LEAST number of Sig Figs.

2.346 X 2.5 =?

5.9. Your answer should be 5.865, then, according to the 2.5 in the problem (the number with the LEAST amount of sig figs) you are now left with 5.8, however, since the digit next to the 8 is ABOVE 5, you round up, leaving you with 5.9 as your FINAL answer.

Rules For Sig Figs AFTER Add/Sub

The decimal places that are left in your final answer are determined by the term with the LEAST amount of Sig Fig Decimal Places. EX: 2.345 + 2.34 + 2.3 = 7.0 Why? Your term with the least amount of decimal places is 2.3, therefore your answer must have the same number of decimal places. When you input this equation on a calc, your answer would be 6.985, but, because of the 8 next to your last digit, 9, you would round up from 6.9 to 7.0. Now why wouldn’t your answer just be 7? As we went over before, 7.0 is more precise than 7, AND 7.0 tells us readers the VALUE of your number.

Rules For Rounding

For rounding the last Sig Digit, you have to look at the digit to the right of it (the first digit that will be dropped). If the first dropped digit is ABOVE 5, you round up. If the digit is smaller than 5, the value does not change. EX: Round 2.367 to 2 sig figs = 2.4. The 6 is the digit to the right of the 3 (the last significant digit) and is above 5, therefore you round your 3 to 4.

In Addition/Subtraction, your Powers of 10

Must be the same before solving. EX: 8.7531 × 10³ + 1.702 × 10^4, you must change the power of 10 in the first problem to 4 to solve. Your new number, according to your new power of ten, would be 0.87531 X 10^4 + 1.702 X 10^4 = 2.57731 X 10^4 (remember that you must round to the term with the LEAST amount of significant digits). Your final answer would be 2.577 X 10^4.

In Multiplication/Division, your Powers of 10

Don’t matter. Remember that in Multiplication and Division you are looking at SIGNIFICANT FIGURES, not digits.