Chapter 1 - Basics of Geometry Section 1.1
... Example 2: Describe the pattern in the sequence of numbers. Predict the next number. a) ...
... Example 2: Describe the pattern in the sequence of numbers. Predict the next number. a) ...
Proofs and Proof Methods
... The Forward-Backward Method and Indirect Proof • Once we have established our premises and our conclusion, we need to construct a chain of equivalences and implications that lead from the premises to the conclusion. To search for this, we might try to find the first step in the chain, by finding so ...
... The Forward-Backward Method and Indirect Proof • Once we have established our premises and our conclusion, we need to construct a chain of equivalences and implications that lead from the premises to the conclusion. To search for this, we might try to find the first step in the chain, by finding so ...
Lecture 10 - 188 200 Discrete Mathematics and Linear Algebra
... Example : Every positive integer is the square of another integer. Proof: The square root of 5 is approximately 2.23 which is not an integer. Hence the statement is false. Q.E.D. ...
... Example : Every positive integer is the square of another integer. Proof: The square root of 5 is approximately 2.23 which is not an integer. Hence the statement is false. Q.E.D. ...
![[Part 1]](http://s1.studyres.com/store/data/008795788_1-6323173b144ce5752d9cfa6a3116d3f8-300x300.png)
Solutions - Full
... Now we see that xy is the product of 4 and an integer (namely, the integer k1 k2 ). So xy is divisible by 4. 6(a) Let a and b be real numbers. Claim 5 |ab| = |a| |b| . Proof. If a = 0 or b = 0, then |ab| = 0 = |a| |b| . Otherwise, there are four cases. 1. If a > 0 and b > 0, then |a| = a and |b| = b ...
... Now we see that xy is the product of 4 and an integer (namely, the integer k1 k2 ). So xy is divisible by 4. 6(a) Let a and b be real numbers. Claim 5 |ab| = |a| |b| . Proof. If a = 0 or b = 0, then |ab| = 0 = |a| |b| . Otherwise, there are four cases. 1. If a > 0 and b > 0, then |a| = a and |b| = b ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".