What is a "significant figure"?

• The number of significant figures in a result is simply the number of figures that are known with some degree of reliability. The number 13.2 is said to have 3 significant figures. The number 13.20 is said to have 4 significant figures

Rules • 1) ALL non-zero numbers (1,2,3,4,5,6,7,8,9) are

ALWAYS significant. • 2) ALL zeroes between non-zero numbers are

ALWAYS significant. • 3) ALL zeroes which are SIMULTANEOUSLY to the

right of the decimal point AND at the end of the number are ALWAYS significant.

• 4) ALL zeroes which are to the left of a written decimal point and are in a number >= 10 are ALWAYS significant.

(1) All nonzero digits are significant:1.234 g has 4 significant figures,1.2 g has 2 significant figures.

(2) Zeroes between nonzero digits are significant:1002 kg has 4 significant figures,3.07 mL has 3 significant figures.

(3) Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point: 0.001 oC has only 1 significant figure,0.012 g has 2 significant figures.

(4) Trailing zeroes that are also to the right of a decimal point in a number are significant: 0.0230 mL has 3 significant figures,0.20 g has 2 significant figures.

5) When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant: 190 miles may be 2 or 3 significant figures,

50,600 calories may be 3, 4, or 5 significant figures. The potential ambiguity in the last rule can be avoided by the use of standard exponential, or "scientific," notation. For example, depending on whether the number of significant figures is 3, 4, or 5, we would write 50,600 calories as:

5.06 × 104 calories (3 significant figures)5.060 × 104 calories (4 significant figures), or5.0600 × 104 calories (5 significant figures).

By writing a number in scientiifc notation, the number of significant figures is clearly indicated by the number of numerical figures in the 'digit' term

as shown by these examples

• A helpful way to check rules 3 and 4 is to write the number in scientific notation. If you can/must get rid of the zeroes, then they are NOT significant.

A helpful way to check rules 3 and 4 is to write the number in scientific notation. If you

can/must get rid of the zeroes, then they are NOT significant.

IF THERE IS NO DECIMAL POINT, THEN TRAILING ZEROES ARE NOT SIGNIFICANT. FOR EXAMPLE:

If there is a decimal point, then all trailing zeroes are significant. For example:

If a number is less than one, then the first significant figure is the first non-zero digit after the decimal point.

WHAT IS AN "EXACT NUMBER"?Some numbers are exact because they are known with complete certainty.Most exact numbers are integers: exactly 12 inches are in a foot, there might be exactly 23 students in a class. Exact numbers are often found as conversion factors or as counts of objects.Exact numbers can be considered to have an infinite number of significant figures. Thus, the number of apparent significant figures in any exact number can be ignored as a limiting factor in determining the number of significant figures in the result of a calculation.

Practice problems• 1) 2804 m • 2) 2.84 km • 3) 0.029 m • 4) 0.003068 m • 5) 4.6 x 105 m • 6) 4.06 x 10-5 m • 7) 750 m • 8) 75 m • 9) 75,000 m • 10) 75,000. m • 11) 75,000.0 m • 12) 10 cm

Scientific Notation

A short-hand way of writinglarge numbers without writing all of the zeros.

An ordinary penny contains about 20,000,000,000,000,000,000,000 atoms. The average size of an atom is about 0.00000003 centimeters across.

The length of these numbers in standard notation makes them awkward to work with.

Scientific notation is a shorthand way of writing such numbers.

The sign of the exponent tells which direction to move the decimal. A positive exponent means move the decimal to the right, and a negative exponent means move the decimal to the left.

Helpful Hint

In scientific notation the number of atoms in a penny is 2.0 1022, and the size of each atom is 3.0 10–8 centimeters across.

2.10,000,000,000,000,000,000,000.

How many decimal places did you move the decimal?

23When the original number is more than 1,

the exponent is positive.The answer in scientific notation is

2.1 x 1023

1) Express 0.0000000902 in scientific notation.

Where would the decimal go to make the number be between 1 and 10?

9.02The decimal was moved how many places?

8When the original number is less than 1, the

exponent is negative.9.02 x 10-8

135,000

1.35 100,000

Think: Move the decimal right 5 places.

A. 1.35 105

1.35 10 5 10 = 100,0005

Additional Example 1A: Translating Scientific Notation to Standard Notation

Write the number in standard notation.

0.0027

Divide by the reciprocal.2.7 100

Think: Move the decimal left 3 places.

2.7 10 –3

B. 2.7 10 –3

10 = –31

100

2.7 1

100

Write the number in standard notation.

Additional Example 1B: Translating Scientific Notation to Standard Notation Continued

Lesson QuizWrite in standard notation.

1. 1.72 104

2. 6.9 10–3

4. 57,000,000

17,200

0.0069

3. 0.0053 5.3 10–3

5.7 107

Write in scientific notation.

5. A human body contains about 5.6 x 106 microliters of blood. Write this number in standard notation.

5,600,000