SIGNIFICANT figures

Two types of numbers: exact and inexact. Exact numbers are obtained by counting or by definitions – a dozen of wine, hundred cents in a dollar

All measured numbers are inexact.

Learning objectives

Define accuracy and precision and distinguish between them

Make measurements to correct precision Determine number of SIGNIFICANT FIGURES in a

number Report results of arithmetic operations to correct

number of significant figures Round numbers to correct number of significant

figures

All analog measurements involve a scale and a pointer

Errors arise from:– Quality of scale– Quality of pointer– Calibration– Ability of reader

ACCURACY and PRECISION

ACCURACY: how closely a number agrees with the correct value

PRECISION: how closely individual measurements agree with one another – repeatability

– Can a number have high precision and low accuracy?

Significant figures are the number of figures believed to be correct

In reading the number the last digit quoted is a best estimate. Conventionally, the last figure is estimated to a tenth of the smallest division

2.0 2.1 2.2 2.3 2.4 2.5

2.3 6

The last figure written is always an estimate

In this example we recorded the measurement to be 2.36

The last figure “6” is our best estimate It is really saying 2.36 ± .01

2.0 2.1 2.2 2.3 2.4 2.5

Precision of measurement (No. of Significant figures) depends on scale – last digit always estimated

Smallest Division = 1 Estimate to 0.1 – tenth of smallest division 3 S.F.99.6

97 98 99 100

Lower precision scale

Smallest Division = 10 Estimate to 1 – tenth of smallest division 2 S.F.96

70 80 90 100

Precision in measurement follows the scale

Smallest Division = 100 Estimate to 10 – tenth of smallest division 1 S.F.90

0 100

Measuring length

What is value of large division?

– Ans: 1 cm What is value of small

division?– Ans: 1 mm

To what decimal place is measurement estimated?

– Ans: 0.1 mm (3.48 cm)

Scale dictates precision

What is length in top figure?

– Ans: 4.6 cm

What is length in middle figure?

– Ans: 4.56 cm

What is length in lower figure?

– Ans: 3.0 cm

Measurement of liquid volumes

The same rules apply for determining precision of measurement

When division is not a single unit (e.g. 0.2 mL) then situation is a little more complex. Estimate to nearest .02 mL – 9.36 ± .02 mL

Reading the volume in a burette

The scale increases downwards, in contrast to graduated cylinder

What is large division?– Ans: 1 mL

What is small division?– Ans: 0.1 mL

RULES OF SIGNIFICANT FIGURES

Nonzero digits are always significant 38.57 (four) 283 (three)

Zeroes are sometimes significant and sometimes not

– Zeroes at the beginning: never significant 0.052 (two)– Zeroes between: always 6.08 (three)– Zeroes at the end after decimal: always 39.0 (three)– Zeroes at the end with no decimal point may or may

not: 23 400 km (three, four, five)

Scientific notation eliminates uncertainty

2.3400 x 104 (five S.F.) 2.340 x 104 (four S.F.) 2.34 x 104 (three S.F.)

23 400. also indicates five S.F. 23 400.0 has six S.F.

Note: significant figures and decimal places are not the same thing

38.57 has four significant figures but two decimal places

283 has three significant figures but no decimal places

0.0012 has two significant figures but four decimal places

A balance always weighs to a fixed number of decimal places. Always record all of them

– As the weight increases, the number of significant figures in the measurement will increase, but the number of decimal places is constant

– 0.0123 g has 3 S.F.; 10.0123 g has 6 S.F.

Significant figure rules

Rule for addition/subtraction: The last digit retained in the sum or difference is determined by the position of the first doubtful digit

37.24 + 10.3 = 47.51002 + 0.23675 = 1002225.618 + 0.23 = 225.85

Position is key

Significant figure rules

Rule for multiplication/division: The product contains the same number of figures as the number containing the least sig figs used to obtain it.

12.34 x 1.23 = 15.1782

= 15.2 to 3 S.F.

0.123/12.34 = 0.0099675850891

= 0.00997 to 3 S.F. Number of S.F. is key

Rounding up or down?

5 or above goes up– 37.45 → 37.5 (3 S.F.)– 123.7089 → 123.71(5 S.F.); 124 (3 S.F.)

< 5 goes down– 37.45 → 37 (2 S.F.)– 123.7089 → 123.7 (4 S.F.)

Scientific notation simplifies large and small numbers

1,000,000 = 1 x 106

0.000 001 = 1 x 10-6

234,000 = 2.34 x 105

0.00234 = 2.34 x 10-3

Multiplying and dividing numbers in scientific notation

(A x 10n)x(B x 10m) = (A x B) x 10n + m

(A x 10n)/(B x 10m) = (A/B) x 10n - m

Adding and subtracting

(A x 10n) + (B x 10n) = (A + B) x 10n

(A x 10n) - (B x 10n) = (A - B) x 10n