Using Numbers in ScienceSignificant Digits

You start by counting the first non-zero digit, and then all those that follow.

Significant digits are all those digits that occupy places for which an actual measurement was taken. These include the last digit which should always be an estimated value

Exceptions

1. If there is no decimal in the number, any zero(s) that the number ends with are not significant.

2. Numbers that are arrived at when counting and exactly defined quantities have an unlimited number of significant digits

Determining the Number of Significant Digits

Examples

1. 5.3451

5 sig. dig.

2. 0.00845

3 sig. Dig.

3. 0.0431000

6 sig. dig.

4. 0.003200

4 sig. dig.

5. 70 000

1 sig. dig.

6. 4.821 x 105

4 sig. dig.

Try These

1. 1.982. 0.028503. 6.5094. 900.005. 8.02 x 10-9

6. 0.000167. 505

3 SD 4 SD 4 SD 5 SD 3 SD 2 SD 3 SD

Scientific NotationWhen writing a number in scientific notation, it is composed of two parts. The mantissa, and the exponent.

2.45 x 106

mantissa exponent

The number is always written so that there is one non-zero digit in front of the decimal point.

Converting from Scientific to Standard Form

If the exponent becomes larger the mantissa must become smaller and if the exponent become smaller the mantissa must become larger to compensate for the change to the exponent. The overall value of the number cannot change.

ie.3.45 x 105 345 0004.285 x 10-3 .0042856.9 x 107 69 000 000

* Keep the same number of SD

Converting Standard Form to Scientific Form

If the mantissa becomes larger the exponent must become smaller and if the mantissa become smaller the exponent must become larger to compensate for the change to the mantissa. The overall value of the number cannot change.

ie.0.000 663 6.63 x 10-4

36 297.6 3.62976 x 104

0.000 000 000439 4.39 x 10-10

Try These

1. 25 3872. 6.92 x 107

3. 0.000 036594. 1.00 x 10-5

5. 88 543 7566. 5.21 x 10-7

7. 3 974 0008. 0.000 000 241

2.538 7 x 104

69 200 0003.659 x 10-5

0.00001008.854 375 6 x 107

0.000 000 5213.974 x 106

2.41 x 10-7

Rounding OffRules

1. If the number is larger than 5, round up.

ie. 52548 = 52550

2. If the number is smaller than 5, round down.

ie. 0.047325 = .047

3. If the number is a 5 followed by all zeroes, or no digits, round to the nearest even number.

ie. 8 775 = 8 780

4. If the number is a 5 followed by any non zero digits, round up.

ie. 0.00445002 = 0.0045

Try These

1) 34 7832) .00217543) 750 0434) 68 5005) 29 3846) 5 550.17) 0.9735

34 800 .002175800 00068 00029 0005 6000.974

Conversionsterra T 1012

giga G 109

mega M 106

kilo k 103

hecta h 102

deca da 101

- 100

deci d 10-1

centi c 10-2

milli m 10-3

micro µ 10-6

nano n 10-9

pico p 10-12

femto f 10-15

Multiply Divide

When doing conversions, if the unit is getting larger, the number must get smaller and if the unit is getting smaller, the number must get larger. However, you must maintain the same number of significant digits.

ie.1000 mm = 1 m The meter is 1 000 times larger than the millimeter, therefore the number must become 1 000 times smaller.

.00462 km = 46.2 dm The dm is 10 000 times smaller than the km, therefore the number is 10 000 times bigger.

ie.

.193 g = 19.3 cg

923 ps = 9.23 x 10-10 s

0.0446 Mm = 4.46 x 105 dm

Try These1) .905 hg = mg2) 822 cm = dm3) 81.4 pm = m4) .00775 g = µg5) 3.76 x 106 kg =

dg

90500 82.2 8.14 x 10-11

7.75 x 103

3.76 x 1010

Multiplying and Dividing

When multiplying or dividing, your answer must be rounded off so that it contains the same number of SD as the value with the least number of SD.

ie. 1. 17 / 42 = .404761904 (must contain 2 SD)

= .40

2. 3.125 x .11 = .34375 (must contain 2 SD) = .34

3. (7.58 x 104) (8.32 x 10-8) / (4.18 x 10-5) = 150.8746411 (3 SD) = 151

Try These1. (35.72) (0.00590) 9. (7.79 x 104) (6.45 x 104)

(5.44 x 106)2. (707000) (3.1)

10. (4.87 x 106) (9.69 x 101)3. (0.05432) (62000) ( 9.765 x 10-3)

4. (6.090 x 10-1) (9.08 x 105) 11. (2.1 x 103) (2.593 x 10-2) (5.23 x 10-3) (6 x 10-5)

5. (1.101 x 109) (4.75 x 109)

6. (6810.12) / (2.4)

7. (.4832) / (5.12)

8. (1.18 x 10-2) / (2.2 x 103)

Answers1) .2112) 2.2 x 106

3) 34004) 5.53 x 105

5) 5.23 x 1018

6) 2800

7) .09448) 5.4 x 10-6

9) 92410) 4.83 x 1010

11) 2 x 108

Adding and SubtractingWhen you add or subtract, the answer must be rounded off to the place value of the least accurate value.

ie. 677

39.2

6.23

722.43 Must be rounded off to the ones place

722 Correct rounded off answer

2) 201 3) 5.32 - 0.75938 = 4.56062

3.57 = 4.56 (hundredths)

98.493

303.063 = 303 (ones)

Try These1) 35.6 + 4.987 + 0.09135

2) 2798 + 33.00 + 45.991

3) .98 - 629.9 + 5300

Answers40.7

2877

4700

Rearranging EquationsTo rearrange an equation for a new variable it must be isolated in the numerator.

Rules

To move a number or variable to the opposite side of an equation, you must perform the opposite operation to that number or variable, and then do the same thing to both sides of the equation.

In order to invert both sides of an equation, you must have a single denominator on both sides of the equation.

𝐴=𝜋 𝑟2 Rearrange for r

1. Divide both sides by

𝐴𝜋

=𝑟2

2. Take the square root of both sides.

√ 𝐴𝜋

=𝑟

Using and Expressing Measurements Measurement – is a quantity that has both a number and a unit.

- it is fundamental to experimental science. - it is important to be able to make measurements and decide if they are correct

Accuracy, Precision and ErrorAccuracy – a measure of how close a measurement is to the actual.

Precision – is a measure of how close a series of measurements are to one another.

Determining ErrorError = experimental value – accepted value

Percent Error = |error|_____ x 100% accepted value