Mathematics Review

Scientific Notation

• Very important for physics class.–Pay attention!

• Allows us to write very large or very small numbers in a more compact form.

Scientific Notation

• Example: 3.75 X 105

• Numbers in scientific notation have three parts:–Coefficient–Base–Exponent

Coefficient BaseExponent

Coefficient

• The coefficient must be between 1 and 9.999999…

• Move the decimal right or left until the coefficient is in the correct range.–Keep only significant digits (more

on this later).

• Remember how many places you moved the decimal!

Base

• We always use 10 as the base in scientific notation.

Exponent• Indicates how many places the decimal

was moved.• To change from scientific to standard

notation, move the decimal:– to the right if the exponent is positive– to the left if the exponent is negative• Like a number line:

• Hint: a number in scientific notation with a negative exponent is a small number, and one with a positive exponent is a big number.

Scientific Notation Practice

• Convert to scientific notation:–125000• 1.25 X 105

–315.2• 3.152 X 102

–0.00634• 6.34 X 10-3

–0.000000774• 7.74 X 10-7

Scientific Notation Practice

• Convert to standard notation:–3.12 X 107

• 31200000

–5.6 X 102

• 560

–7.41 X 10-4

• 0.000741

–9.34 X 10-8

• 0.0000000934

Significant Digits

• To be honest: Your understanding of significant digits will not be tested on the Physics Regents…

• …But I think you should understand it anyway!

• There are 4 rules for significant digits.

Significant Digits

• Rule #1• ALL non-zero digits are

significant (they count).

• 934 (3 significant digits)• 227538 (6 significant digits)

Significant Digits

• Rule #2• All zeroes between non-zero

digits are significant (they count)

• 307 (3 significant digits)• 10108 (5 significant digits)

Significant Digits• Rule #3• Trailing zeros (those at the end)

are significant only if the number contains a decimal point; otherwise they are insignificant (they don’t count).

• 1.34500 (6 significant digits)• 650.0 (4 significant digits)• 650 (2 significant digits)

Significant Digits

• Rule #4• Zeros to left of the first nonzero digit

are insignificant (they don’t count); they are only placeholders!

• 0.563 (3 significant digits)• 0.0563 (3 significant digits)• 0.0000000000000000000563 (3

significant digits)

Significant Digits Practice

• How many significant digits?–120700• 4

–0.00318• 3

–9.00340• 6

–1.0000• 5

Algebraic Solutions

• Sometimes it is useful to solve an equation for the important variable BEFORE you plug in the numbers.

• There aren’t specific rules for this, you just need to practice.

Algebraic Solutions Practice

• Solve for “x” in terms of “a” “b” and “c”:–c = x + 3• x = c - 3

–b = ax• x = b/a

–a = bx +c• x = (a – c)/b

–b3 = 2x2 + c• x = √((b3 – c)/2)

Measuring

• Metric Ruler / Meter Stick– NEVER FEET & INCHES!!!

• 1 meter = 100 cm = 1000 mm• 1 cm = 10 mm

Measuring• When using a meter stick, 1 stick = 1

meter (duh).• On metric rulers or meter sticks the long

lines mark centimeters and the short lines mark millimeters.– Usually the line marking every 5th

millimeter is slightly longer.

CentimeterMillimeter 5th Millimeter

Measuring• Angles• Line up your protractor as shown.

• Use your knowledge of geometry to decide which scale to use.– For acute angles use the smaller number;

for obtuse angles use the larger number.

Graphing• How to graph:– 1. Review your

data.

Distance (m)

Time (s)

1 1

3 2

4 3

6 4

8 5

Graphing• How to graph:– 2. Label axes and scales.

• Include units in axis label.

• Always start scale at zero.

• Time always goes on the x-axis

Distance (m)

Time (s)

1 1

3 2

4 3

6 4

8 5Time (s)

Dis

tan

ce (

m)

0 1 2 3 4 5 6 7 8

8 7 6 5 4 3 2 1

Graphing• How to graph:– 3. Plot data

Distance (m)

Time (s)

1 1

3 2

4 3

6 4

8 5Time (s)

Dis

tan

ce (

m)

0 1 2 3 4 5 6 7 8

8 7 6 5 4 3 2 1

Graphing• How to graph:

– 4. Draw a line of best fit.• Do NOT connect the

dots!• One line that comes

close to all data points.

Distance (m)

Time (s)

1 1

3 2

4 3

6 4

8 5 Time (s)

Dis

tan

ce (

m)

0 1 2 3 4 5 6 7 8

8 7 6 5 4 3 2 1

Graphing• How to graph:

– 5. If asked, find slope.• Pick two points on

best fit line.• Do NOT pick data

points unless they actually fall on the best line.

• Slope = ¢y / ¢x• We’ll use (1.5s, 2m)

and (3.5s, 5m).• Slope =

(5m-2m)/(3.5s-1.5s)• Slope = 3m / 2s =

1.5 m/sTime (s)

Dis

tan

ce (

m)

0 1 2 3 4 5 6 7 8

8 7 6 5 4 3 2 1