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: 3600.007 (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: 3.6000000 (significant zeros) 3600.00000 (significant zeros) No other zeros are significant. Examples: 36000000. (NOT significant zeros) .00036 (NOT significant zeros)
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.
4	Examples of Numerical Accuracy 3.6 has 2 place accuracy. (2 significant digits) 30.607 has 5 place accuracy. (5 significant digits) (3.6)(30.607) = 110.1852, but this is accurate only to two places because of the accuracy of the 3.6. The answer should be given as 110, rounded. p » 3.14159 (6 significant digits); 2 is an exact number; and r = 1.25 (3 significant digits) 2pr = 7.78975, but gets reported as 7.79, rounded. p ¸ 2 = 1.570795, but gets reported as 1.57080, rounded.
5	Example of Precision 527.03 (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. 527.03 + 120 = 647.03, 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.
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”.
7	Some Common Numbers the velocity of light is about 299,790,000 meters/sec the charge on an electron is about 0.000 000 000 000 000 000 160 2 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 0.000 000 000 1 meter. Such large and small numbers are extremely difficult to use in this form. A more convenient notation is needed …
8	Scientific Notation 299,790,000 meters/sec is written: 2.9979 x 10⁸ 0.000 000 000 000 000 000 160 2 coulomb is written: 1.602 x 10^–19 602,213,670,000,000,000,000,000 is written: 6.022 x 10²³ 0.000 000 000 1 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.
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 = 330,000. (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 = 0.00033 (3.3 > .00033) Make the exponent positive when the number as given must be made smaller (decimal point moved left): 2,015,000. = 2.015 x 10⁶ (2,015,000 > 2.015) Make the exponent negative when the number as given must be made larger (decimal point moved right): 0.000 201 5 = 2.015 x 10 ^–4 (0.000 201 5 < 2.015)
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:
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)
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 ¹ 0 You first collect together all terms containing the variable. It is sometimes easier to eliminate denominators first.
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
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)
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
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).