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.
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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
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”).
Real number = integer-part + fractional-part
Surds: an irrational root such as √3
lacking sense : IRRATIONAL; absurd
4 + 2i
all have to be either -2 or +2
Radicals become easier if you think of them in terms of indices. Think instead of
Rational Numbers VS. Irrational Number
100.3; 1/6 = .16666; 2/7 = .285714285714
Number that can’t be expressed as p/q. Not a quotient of two integers
2½ = 1.4142135623730950488016887242097….
Approximating irrational number by rational numbers: number theory
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).
1 x 25 = 1 x 5 x 5 5 + 5 = 10
16 x 1 = 4 x 4 4 + 4 = 8
Page 30 example 12, 13
1 x 12 = 3 x 4 3 + 4 = 7
2 x 15 = 2 x 3 x 5 = 6 x 5 6 - 5 = 1
Page 39 example 6
Page 42 – Example 10
Combine the numerator terms
On the 2nd day You climbed 4 miles vertically
On the 2nd day, You covered 3 miles horizontally
3rd day end point
2nd day start point
How far did we walk ?
Equation of a straight line
Now that we know there is a right-triangle, how far did we walk ?
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
We can find AB using the Pythagoras’ theorem
Mid-point of line segment AB
x and y are variables -- various points along the line
Slope of a line joining points (0,c) and (x,y)
x can’t be 0
Point (0,c) lies on y axis
Set y = 0 to find the x-intercept
Set x = 0 to find the y-intercept
When m = 0
no incline, line is parallel to x-axis. No x intercept
y = c
M can’t be 0
y = c
No gradient (undefined), straight up, perpendicular to the x axis. No y intercept. Parallel to y axis. x = k.
c is y-intercept
Ex: (1,2), (-1,2), (5,2)…
Example: given gradient, and a point on the line, find the line’s equation
Slope of the line is given
The lines passes through (2,1)
Try this: (-2,3); m = -1 y = -x + 1
Example: given two points on a line, find the line’s equation
Step2: once m is known, use the same equation and one of the points to find the equation
Step1: given two points, it is easy to find m
Try this: (3,4), (-1,2) 2y = x + 5