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In the early days, one orange, 3 oranges, ..etc. any of the natural numbers, the negatives of these numbers, or zero natural numbers (a positive integer) Root, Radix, Radicals Real Number VS. Imaginary (complex) Number Real number = integer-part + fractional-part Ninth-century Arab writers called one of the equal factors of a number a root, and their medieval translators used the Latin word radix (“root,” adjective “radical”). 4 1 2 2 2 1 2i 2 2 22 4 4 not real since - 4 - 2 -2 or 2 2 4 + 2i real part 4 2 2 all have to be either -2 or +2 imaginary part 2 2 1 Surds: an irrational root such as √3 2 2 fractional index lacking sense : IRRATIONAL; absurd Radicals become easier if you think of them in 1 terms of indices. Think 4 2 instead of 4 http://www.itc.csmd.edu/tec/GGobi/index.htm Rational Numbers VS. Irrational Number 100.3; 1/6 = .16666; 2/7 = .285714285714 Approximating irrational number by rational numbers: number theory Number that can’t be expressed as p/q. Not a quotient of two integers 2½ = 1.4142135623730950488016887242097…. (3.1416…) http://mathworld.wolfram.com/Pi.html How do you represent large multiples such as 2x2x2x2 takes too much space to print 2x2x2x2 = 24 the birth of exponential notation (base, exponent or index (indices)) Now we need a set of rules to figure out what things such as is 22 x 23 Or 23 x 32 Properties of exponents Logarithms: Math based on the exponents themselves, invented in the early 17th century to speed up calculations. Also from the result of the study of arithmetic and geometric series. (study tip: the exponent is the logarithm). integer any of the natural numbers, the negatives of these numbers, or zero natural numbers the number 1 or any number (as 3, 12, 432) obtained by adding 1 to it one or more times : a positive integer real number one of the numbers that have no imaginary parts and comprise the rationals and the irrationals imaginary number a complex number (as 2 + 3i) in which the coefficient of the imaginary unit is not zero irrational number a number that can be expressed as an infinite decimal with no set of consecutive digits repeating itself indefinitely and that cannot be expressed as the quotient of two integers Rational Numbers an integer or the quotient of an integer divided by a nonzero integer Radical Root; Foundation Page 18 – Example 10b 3 16x 3 54x 4 16 2 8 2 2 2 2 2 2 x x3 3 54 2 27 2 33 2 2 x 3 1 3 2 33 x x 3 22 x 3 3x2 x 1 2 x 3 2 3x 3 2 x 2 3x 1 1 3 1 3 8 x 9 = 4 x 2 x 9 = 4 x 18 18 – 4 = 14 8 x 2 14 x 9 8 x 2 (18 4) x 9 8 x 2 18 x 4 x 9 8 x 2 4 x 18 x 9 4 x(2 x 1) 9(2 x 1) (2 x 1)( 4 x 9) Page 29 example 9 1 x 25 = 1 x 5 x 5 5 + 5 = 10 16 x 1 = 4 x 4 4 + 4 = 8 x 2 10 x 25 x (5 5) x 25 2 x 2 5 x 5 x 25 xx 5 5x 5 x 5x 5 x 52 16 x 2 8 x 1 16 x 2 (4 4) x 1 16 x 2 4 x 4 x 1 4 x4 x 1 1(4 x 1) 4 x 14 x 1 4 x 12 Page 30 example 12, 13 1 x 12 = 3 x 4 3 + 4 = 7 x 2 7 x 12 x 2 3 4x 12 x 2 3x 4 x 12 xx 3 4x 3 x 3x 4 2 x 15 = 2 x 3 x 5 = 6 x 5 6 - 5 = 1 2 x 2 x 15 2 x 2 (6 5) x 15 2 x 2 6 x 5 x 15 2 x x 3 5 x 3 x 32 x 5 Page 40 example 7 Page 39 example 6 x 2 x 3 3x 4 x3 x 4 2 x 3 x 33x 4 3x 2 4 x 2 x 6 x 33x 4 3x 2 2 x 6 x 33x 4 3 2 x3 x 1x 1 2 x 1 x x 1 3x x 1 2x 1x 1 x 3x x 1x x 1 3x 2 3x 2 x 2 2 x 2 3x x 1x x 1 2x2 6x 2 x 1x x 1 Page 42 – Example 9 x1 2 x 2 1 2 x 2 x 1 1 1 1 2 x 1 1 2 x 2 1 2 x 2 3 1 3 1 2 2 1 1 2 x 2 1 1 1 2 x 2 1 1 4 x 2 2 x 1 1 2 x 2 2 1 2 2 1 2 2 x 11 2 x 1 2 x Page 42 – Example 10 x2 4 x 4 x 4 x 4 x 4 x x 4 x 1 1 2 2 x2 Combine the numerator terms 2 2 2 1 2 4 4 x 2 1 x 1 2 x 2 1 1 x2 4 x 2 1 2 4 x 3 2 a b a c bc 1 2 x2 4 x2 4 x2 4 4 x 4 x 2 1 2 2 4 4 x 2 3 2 1 1 3 2 2 2 1 2 Climbing the mountain on a straight slope 8 10, 7 7 6 On the 2nd day You climbed 4 miles vertically 5 4 4, 3 3 2 1 0 0 2nd 2 day start point 4 6 8 10 On the 2nd day, You covered 3 miles horizontally 12 3rd day end point y2 y1 vertical height 73 4 2 gradient ( slope ) horizonal height 10 4 6 3 x2 x1 How far did we walk ? Slope of a Straight Line tangent y2 y1 m x2 x1 Equation of a straight line • We need to know two points (locations) – The second day’s starting location and ending location • Can you identify the right-triangle in the previous slide? • Can you identify the right-angle? • It is customary to denote the slope of a straight line by “m” Now that we know there is a right-triangle, how far did we walk ? Given two point on a line, what is the distance between the two points? 8 B 7 Vertical distance 10, 7 6 5 4 A 4 3 C (4, 3) 2 (10,3) 6 1 0 0 2 4 6 8 10 12 We were only given points A and B. Using A and B we could simply figure out point C. Point C is same height as point A but it is (10 – 4) or 6 units away from A Horizontal distance We can find AB using the Pythagoras’ theorem AB (10 4) 2 (7 3) 2 6 2 4 2 AB ( x2 x1 ) 2 ( y2 y1 ) 2 1 1 x x , y y 2 1 2 Mid-point of line segment AB 2 1 2 Equation of a line y2 y1 m y2 y1 mx2 x1 y2 mx2 x1 y1 x2 x1 y mx c slope x and y are variables -- various points along the line Slope of a line joining points (0,c) and (x,y) y-intercept yc m x0 x can’t be 0 Point (0,c) lies on y axis Properties of y mx c Set y = 0 to find the x-intercept Set x = 0 to find the y-intercept When m = 0 x c m no incline, line is parallel to x-axis. No x intercept M can’t be 0 No gradient (undefined), straight up, perpendicular to the x axis. No y intercept. Parallel to y axis. x = k. y = c y=c c is y-intercept Ex: (1,2), (-1,2), (5,2)… 2.5 5 4 2 (-1, 2) 1.5 (3, 4) 3 2 (5, 2) (2, 2) 1 (3, 2) 0.5 1 (3, 0) 0 0 1 2 3 4 0 -2 0 2 4 6 Typical problems involving straight lines? • Find whether 4 points form a parallelogram. – Method1: Calculate the distances between them to see if AB = DC and CB = DA – Method2: Using mid-points • If the mid-points of the diagonals AC and BD bisect each other then ABCD is a parallelogram – Mehtod3: Using gradients • If the gradients of AB and DC are same 4 3 B(5, 3) 2 A(1, 1) 1 C(3, 0) 0 -2 0 2 4 -1 D(-1, -2) -1,-2-2 -3 Matlab: plot([-1,1,5,3,-1],[-2,1,3,0,-2]) 6 Example: given gradient, and a point on the line, find the line’s equation Slope of the line is given P(x,y) m y2 y1 x2 2 x2 x1 y 1 The lines passes through (2,1) y=2x-3 A(2,1) Try this: (-2,3); m = -1 y = -x + 1 Example: given two points on a line, find the line’s equation Step1: given two points, it is easy to find m m y2 y1 x2 x1 Step2: once m is known, use the same equation and one of the points to find the equation m y2 y1 x2 x1 Try this: (3,4), (-1,2) 2y = x + 5