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MATH 103

by: Ariel Harris

MATH 103 Math 103

Ariel Harris

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These notes go over fractions/rational numbers. They also introduce the notion that infinty is an almost impossible concept.
Debra Warne
Class Notes
math jmu week5 natureofmathematics
25 ?





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This 7 page Class Notes was uploaded by Ariel Harris on Thursday February 11, 2016. The Class Notes belongs to Math 103 at James Madison University taught by Debra Warne in Spring 2016. Since its upload, it has received 12 views. For similar materials see THE NATURE OF MATHEMATICS in General at James Madison University.


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Date Created: 02/11/16
CLASS NOTES: February 8 2016 More rational numbers p { q | p, q ∈ ℤ, where q ≠ 0} Rational number symbol: ℚ (Thinking of the term Quotient) Example from reading (first page of chapter 5) A quarter of a quarter (of a whole)  "of a whole" is implied, but not written. We first need to look at the quarter of the whole. The best way to show a whole is to use a circle because it is easier to cut into multiple pieced evenly. This circle indicates a whole this indicates the whole in quarters. The sentence is asking for a quarter of a quarter. That indicates one fourth of the whole. 1/4 of 1/4 of 1 Now we need to cut the fourth into fourths.  Therefore, the whole piece would look like this (except a lot better Because I am really bad at shapes in word) 1/4 of 1/4 = 1/16 aka: 1/4 * 1/4= 1/16 Recall: fraction multiplication a * p = b q Where a,b,p,q are ∈ℤ and b,q are ≠0 a p a∗p b * q = b∗q Another example: one third of a half (of a whole) – ( …) implied Half of this circle represents the half we are focusing on. We now need to separate the half into thirds. 1 ∗1 1 of1= = 3 3 2 6 2 Are rational numbers closed under multiplication? a b∗p ap = } These integers are closed under multiplication q bq Yes, the rational numbers are closed under multiplication, addition, and subtraction. 2 1 − Example: 3 4  Find a common denominator. The common denominator is known as re-writing each number in a more convenient form from its infinitely many ways to be written so that we're putting each into the same number of pieces. - = 2∗4 2 3 8 3= 4 = 12 2 1 ∗4 ∗3 2− = 3 − 4 3 4 4 3 8 3 5 ¿12 −12 =12 The general abstract version looks like this for addition: a,b,p,q,∈ ℤ, b, q ≠0 a p b q a p ∗q ∗b ¿ b + q q b aq pb ap+pb ¿ bq qb= qb Note that bq = qb by commutative law of multiplication for integers Rational numbers have been around for thousands and thousands of years. We also become aware of decimal representations for rational numbers. 1 Example: 4=¿ 0.25 (terminating decimal) 1 Example: = 3 0.3333333333333333333333333333333333333333333 1  3 = 0. 3 The over bar tells us to repeat this forever. The definition of rational numbers can be used to prove that every rational number must have a decimal representation that either terminates or has a repeating final pattern that goes on forever. Class notes: p Rational numbers: ℚ { q∨¿ p,q ∈ ℤ and q ≠ 0 Question: are the rational numbers closed under division? n Closed under division: given any two n, m, ∈ ℚm must also ∈ ℚ Answer: No, since 0 ∈ ℚ, we cannot divide by 0 and still get a rational number. Counter Example: 1 0 Let n = so n ∈ ℚ Since 1, 2 ∈ ℤ and 2 ≠ 0, let m = so m ∈ ℚ since 2 1 0,1 ∈ ℤ and 1 ≠ 0 n But, m = 0 so dividing m ∉ ℚ since m=0 Sets are closed under operation, but operations are not closed. We could tweak ℚ: consider ℚ - {0} Then this represents the set of all rational numbers excluding 0. Then the rational numbers are closed under division. a b c } a, b, c, d ∈ ℤ b, d ≠ 0 and c ≠0 d Last time we noted that while rational numbers have been around for many thousands of years and the decimal system is more of a modern day development (in the last couple hundred years) calculators have caused us to think about decimal expansions of rational numbers. So let's soldify some ideas there! 0.25 4 1.000 0.8 1 0.20 4 0.20 0.00 = 0.25 Why? Long division! The decimal expansion terminates (the long division finally reaches zero.) 1 3 3 = 0. Why? Long Division! (long division leads to realizing that the decimal will never stop. The decimal places repeat forever. It can be proven that every rational number must either have a terminating decimal representation or settle into a specific repeating patter of numbers that can be expressed exactly by using the over bar notation. 1 Let's do the computation in the book: 7 7 divided by 1 = 0.142857 142857 1 ´ ∴ we see why 7 = 0. 142857 Now let's look at more mind-blowing ideas. 1 Back to 3 1 1 1 Do you agree that + + =1 ? 3 3 3 1+1 = + = =13 3 3 3 3 1 3 But we also just showed that 3 is just the same as 0. . Just to make it 1 more explicit 3 =0.3333333333333333… What if we do: 0.3 3 3 3 3 3 3 3 3 3 0.3 3 3 3 3 3 3 3 3 3 + 0.3 3 3 3 3 3 3 3 3 3 0.9 9 9 9 9 9 9 9 9 9 So now we've learned that 1=0. 9


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