Chapter 1
... 1.1 Properties of Real Numbers Real numbers – any # you can think of. From -∞ to +∞ Real Numbers Irrational Numbers ...
... 1.1 Properties of Real Numbers Real numbers – any # you can think of. From -∞ to +∞ Real Numbers Irrational Numbers ...
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... Theorem I.For any positive integer/, in order that y be a Lucas number, it is necessary and sufficient that there exist a positive number x such thait ...
... Theorem I.For any positive integer/, in order that y be a Lucas number, it is necessary and sufficient that there exist a positive number x such thait ...
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... All the equations (i)-(vi) involve contradictions. Of these, perhaps (ii) is the least obvious. Let us therefore examine (ii), which is true for m = 2 (even) leading to c2 = 1, cx = 2 from (ii) and (8). Now c2 = 1 = a2 - h2 implies that a2 = 2 (b2 = 1) or a2 - 1 (b2 = 0), i.e., a2 ^ 0, which contrad ...
... All the equations (i)-(vi) involve contradictions. Of these, perhaps (ii) is the least obvious. Let us therefore examine (ii), which is true for m = 2 (even) leading to c2 = 1, cx = 2 from (ii) and (8). Now c2 = 1 = a2 - h2 implies that a2 = 2 (b2 = 1) or a2 - 1 (b2 = 0), i.e., a2 ^ 0, which contrad ...
Lecture 2: Irrational numbers
... We want to appreciate one of the great moments of mathematics: the insight that there are numbers which are irrational. It was the Pythagoreans, who realized this first and - according to legend - tried even to ”cover the discovery up” and kill Hippasus, one of the earlier discoverers. We have seen ...
... We want to appreciate one of the great moments of mathematics: the insight that there are numbers which are irrational. It was the Pythagoreans, who realized this first and - according to legend - tried even to ”cover the discovery up” and kill Hippasus, one of the earlier discoverers. We have seen ...
Extending the Number Line
... number line as the original number, but on the other side of 0. Zero is its own opposite. Integers are the set of whole numbers and their opposites ...
... number line as the original number, but on the other side of 0. Zero is its own opposite. Integers are the set of whole numbers and their opposites ...
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... Note that in (2) the base numbers are distinct except perhaps for ql=q0. We shall show that when this happens either c0 = 0 or else cx = 0; that is, the base number 1 occurs at most once in each evaluation of (3). For a proof, suppose that the proposition is false for some r, and let n be the least ...
... Note that in (2) the base numbers are distinct except perhaps for ql=q0. We shall show that when this happens either c0 = 0 or else cx = 0; that is, the base number 1 occurs at most once in each evaluation of (3). For a proof, suppose that the proposition is false for some r, and let n be the least ...
1-8 Number Lines
... Negative Integers + Positive Integers + all the stuff in between = Real Numbers (R) ...
... Negative Integers + Positive Integers + all the stuff in between = Real Numbers (R) ...
![[Part 1]](http://s1.studyres.com/store/data/008795780_1-2d32093eb8955eecb4220b99bc38a981-300x300.png)
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... J. L.BROWN, JR. and R. L DUWCAN The Pennsylvania State University, University Park, Pennsylvania ...
... J. L.BROWN, JR. and R. L DUWCAN The Pennsylvania State University, University Park, Pennsylvania ...
1.1 B Sets and Real Numbers February 07, 2011
... Understand the set of real numbers and the subsets of real numbers. Order numbers on the real number line. Determine the distance between two numbers on the real number line. Determine the absolute value of a real number. ...
... Understand the set of real numbers and the subsets of real numbers. Order numbers on the real number line. Determine the distance between two numbers on the real number line. Determine the absolute value of a real number. ...
