1. Precision, Coordinate Plane, and Pythagoras

2. Significant Digits All nonzero digits in a number are significant.
All zeros between two nonzero digits are significant.
Example: (significant zeros)
A string of zeros after (to the right of) significant digits are significant ONLY if they end to the right of the decimal point.
Examples: (significant zeros)
(significant zeros)
No other zeros are significant.
Examples: (NOT significant zeros)
(NOT significant zeros)
Only zeros are a problem. Are they significant or not?
1/2/2019
Dave Saha

3. Numerical Accuracy
The accuracy of a number is the number of its significant digits.
The precision of a number is the digit position of its least (rightmost) significant digit.
An exact number is always more accurate than any approximate number, no matter what the accuracy of the approximate number.
The accuracy of a product or quotient is equal to the least accurate factor in the multiplication or division.
The precision of a sum or difference is equal to the highest order of any term’s least significant digit.
1/2/2019
Dave Saha

4. Examples of Numerical Accuracy
3.6 has 2 place accuracy. (2 significant digits)
has 5 place accuracy. (5 significant digits)
(3.6)(30.607) = , but this is accurate only to two places because of the accuracy of the The answer should be given as 110, rounded.
 (6 significant digits); 2 is an exact number; and r = 1.25 (3 significant digits) 2r = , but gets reported as 7.79, rounded.
  2 = , but gets reported as , rounded.
1/2/2019
Dave Saha

5. Example of Precision
(5 significant digits) is precise to the hundredths position. Its least significant digit is the 3 in the hundredths position.
120 (2 significant digits) is precise to the tens position. Its least significant digit is the 2 in the tens position.
= , but the last (rightmost) digit with total accuracy is the tens digit. The lack of an accurate unit’s digit in 120 affects the accuracy of the sum. So we report the sum as 650, rounded.
1/2/2019
Dave Saha

6. Writing Numbers
Our way of writing numbers is called a positional system.
For example, in 220.2, the leftmost 2 is different from the middle 2, and different again from the rightmost 2.
The difference is the size of the “thing” indicated by each digit. The leftmost 2 counts “hundreds”; the middle 2 counts “tens”; the rightmost 2 counts “tenths”.
Notice, also, that the the zero indicates (or “counts”) that there are no “ones”.
1/2/2019
Dave Saha

7. Some Common Numbers
the velocity of light is about 299,790,000 meters/sec
the charge on an electron is about coulomb.
the number of hydrogen atoms that weigh 1 gram is about 602,213,670,000,000,000,000,000.
the size of atoms is on the scale of meter.
Such large and small numbers are extremely difficult to use in this form. A more convenient notation is needed …
1/2/2019
Dave Saha

8. Scientific Notation 299,790,000 meters/sec is written: 2.9979 x 108
coulomb is written: x 10–19
602,213,670,000,000,000,000,000 is written: x 1023
meter is written: 1. x 10–10
The exponent of the 10 informs us how many positions and in which direction the decimal point, as given, needs to be shifted to place the leading significant digit where it should be.
1/2/2019
Dave Saha

9. Practice with Scientific Notation
When the exponent is positive, the number as given must be made larger (move the decimal point right):
3.3 x 105 = 330, (3.3 < 330,000)
When the exponent is negative, the number as given must be made smaller (move the decimal point left):
3.3 x 10–4 = (3.3 > )
Make the exponent positive when the number as given must be made smaller (decimal point moved left):
2,015,000. = x 106 (2,015,000 > 2.015)
Make the exponent negative when the number as given must be made larger (decimal point moved right):
= x 10 –4 ( < 2.015)
1/2/2019
Dave Saha

10. Scientific Notation and Your Calculator
Your calculator will accept input of numbers in modified form of scientific notation. Instead of pressing keys for x10^ , all you have to do is press the e key. (Some calculators label this key EE .)
Examples:
the speed of light, approx. 3.0 x 108, can be entered as 3 e 8
the elementary charge on an electron, –1.602 x 10–19, can be entered as –1.602 e –19
1/2/2019
Dave Saha

11. Properties of Linear Equations
Remember these basic rules for addition and multiplication :
associative property of addition (a + b) + c = a + (b + c)
commutative property of addition a + b = b + a
associative property of multiplication [(a)(b)](c) = (a)[(b)(c)]
commutative property of multiplication (a)(b) = (b)(a)
1/2/2019
Dave Saha

12. Properties of Linear Equations
distributive property (a)(b + c) = (a)(b) + (a)(c)
if A = B, then A + C = B + C
if A = B, then A – C = B – C
if A = B, then (A)(C) = (B)(C)
if A = B, then A/C = B/C, for C  0
You first collect together all terms containing the variable.
It is sometimes easier to eliminate denominators first.
1/2/2019
Dave Saha

13. Linear Equation Example
5(t – 4) = –(7 – 2t)
5t – 20 = –(7 – 2t)
5t – 20 = –7 + 2t
3t – 20 = –7
3t = 13
t =
Given
Distributive property on the left
Distributive property on the right
Subtract 2t from both sides
Add 20 to both sides
Divide by 3 on both sides
1/2/2019
Dave Saha

14. Cartesian Coordinates
The x-coordinate, which indicates how far left or right of the y-axis is the point, is the first number of the ordered pair: (x, )
The y-coordinate, which indicates how far up or down from the x-axis is the point, is the second number of the ordered pair: ( , y)
1/2/2019
Dave Saha

15. The Pythagorean Theorem
Right triangles occur very often, so the Pythagorean theorem is essential to know: a2 + b2 = c2
Common Pythagorean triangles with integer sides:
3, 4, 5 5, 12, 13 8, 15, 17
32 = 9
+42 = 16
52 = 25
52 = 25
+122 = 144
132 = 169
82 = 64
+152 = 225
172 = 289
1/2/2019
Dave Saha

16. The Distance Formula
Think of the distance formula as the length of the radius from one point (as center of a circle) to another point (on the circle itself).
The distance formula is a special case of the Pythagorean Theorem.
(x2, y2)
(x1, y1)
1/2/2019
Dave Saha