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1.1) Real number
- Any number that can be expressed in decimal form.
- For example: 7 (=7.000...), √2 (=1.4142...), -2/3 (= -0.6666667)
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How do you indicate that a decimal repeats indefinitely?
Writing a bar above it
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Real numbers can be represented as points on a ______ ____. The point associated with the number zero is referred to as the _____.
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There is a one-to-one correspondence between the set of real numbers and the set of points on the line. Explain
This means that each real number is identified with exactly one point on the line; conversely, with each point on the line we identify exactly one real number.
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The real number associated with a given point is called the ______ of the point
coordinate
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Natural numbers
ordinary counting numbers, like 1,2, &3
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Integers
Natural numbers along with their negatives and zero
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Rational numbers
Real numbers that are ratios of two integers with nonzero denominators
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It can be proved that a real number is rational if and only if its decimal expansion ________ or _______
terminates (3.15) or repeats (2.433333...)
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Irrational numbers
Real numbers that are not rational for example √2 and π (pi)
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Intervals (define) can be characterized via _______
- uninterrupted portion of the number line
- inequalities
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Open interval (a,b)
consists of all real numbers x such a<x<b
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Closed interval [a,b]
- consists of all real numbers x such that a≤x≤b
- the brackets [] indicate the numbers a and b are included in the interval
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unbounded intervals
intervals that extend indefinitely in one direction or the other for example (2,∞)
*keep in mind ∞ is not a real number, it is just used to indicate that the interval has no right hand boundary
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(-∞,∞)
the entire real-number line
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Any number with the form √n where n is not a ______ _____ is irrational. Also any sum, difference, product and quotient of an irrational number and a nonzero rational are all ________
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1.2) Absolute Value (geometric and algebraic definition)
- geometric: absolute value of a real number |x|, is the distance from x to the origin
- algebraic: absolute value of a real number |x|, is x itself when x≥0 and the value of |x| is -x when x <0.
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For real numbers a and b, the distance between them is?
|a-b| =|b-a|
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1.3) A variable can be part of a given set of numbers called the ______.
domain
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Domain of a variable (in a given expression)
the set of all real-number values of the variable for which the expression is defined
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A linear or ____-_____ equation in one variable is an equation that can be written in the form:
- first-degree equation in one variable
- ax + b = 0
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When you attempt to check an answer to prove the equation was accurately solved and it comes up undefined what does that mean?
The equation has no solution, in this case the answer derived is called an extraneous root or an extraneous solution
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1.4) Formula for area and circumference of a circle
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Midpoint formula
[(x1+x2)/2, (y1+y2)/2]
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Percent error formula
percentage error = [|(actual value)-(approx. value)|/actual value ]*100
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2.3) If two inequalities have the same problem set, they are said to be ______. If you multiply or divide both sides of an inequality by a negative number, you must ______ the inequality
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Steps for Solving Polynomial Inequalities (5)
- If necessary, rewrite the inequality so that the polynomial is on the left-hand side and zero is on the right-hand side
- Find the key numbers for the inequality and locate them on a number line
- List the intervals determined by the key numbers
- From each interval, choose a convenient test number. Then use the test number to determine the sign of the polynomial throughout the interval
- Use the information obtained in the previous step to specify the required solution set [Don't forget to take into account whether the original inequality is strict (< or >) or nonstrict (≤ or ≥).]
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