Chapter 1-sec1.1
... The contrapositive of this statement is “if x is even, then x2 is even,” which is true by Example 5. QED. Example 7 If x2 is even then x is even. Proof: This is the contrapositive of Example 4, which has been shown to be true. QED. If And Only If (Iff) Proofs A statement of the form “A if and only i ...
... The contrapositive of this statement is “if x is even, then x2 is even,” which is true by Example 5. QED. Example 7 If x2 is even then x is even. Proof: This is the contrapositive of Example 4, which has been shown to be true. QED. If And Only If (Iff) Proofs A statement of the form “A if and only i ...
section 1.8 words into symbols
... KEY WORDS / PHRASES TO LOOK FOR: 1.) ADDITION – plus, sum of, increased by, more than 2.) SUBTRACTION – minus, difference, decreased by, less than 3.) MULTIPLICATION – product of, times, of 4.) DIVISION – divided by, quotient of 5.) EQUALS – is, is equal to, are, was, will be - use this to help you ...
... KEY WORDS / PHRASES TO LOOK FOR: 1.) ADDITION – plus, sum of, increased by, more than 2.) SUBTRACTION – minus, difference, decreased by, less than 3.) MULTIPLICATION – product of, times, of 4.) DIVISION – divided by, quotient of 5.) EQUALS – is, is equal to, are, was, will be - use this to help you ...
A Fibonacci-like Sequence of Composite Numbers
... (It is easy to check that the second property above holds, because mk is the first subscript such that Fmikis divisible by Pk The third property holds because the first column nicely "covers" all odd values of n; the middle column covers all even n that are not divisible by 6; the third column cover ...
... (It is easy to check that the second property above holds, because mk is the first subscript such that Fmikis divisible by Pk The third property holds because the first column nicely "covers" all odd values of n; the middle column covers all even n that are not divisible by 6; the third column cover ...
The Value of the Number Line
... MCC6.NS.5: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. MCC6.NS.6a: Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a ...
... MCC6.NS.5: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. MCC6.NS.6a: Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a ...
Full text
... This note describes an alternative approach to the proof in [2] of a representation theorem involving negatively subscripted Pell numbers P_n (w > 0), namely, Theorem: The representation of any integer N as ...
... This note describes an alternative approach to the proof in [2] of a representation theorem involving negatively subscripted Pell numbers P_n (w > 0), namely, Theorem: The representation of any integer N as ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.