Math 120 Midterm 1
There are other non-algebraic numbers called...
Transcendental numbers, including pi, e, and log(2)
True or false? Pythagoreans believed all numbers to be rational.
True. A reject named Hippasus proved that root 2 is irrational. No one knows what happened to him
Complete the life lesson: Don't overlook or dismiss facts...
that seem insignificant or irrelevant
Complete the life lesson: A rock of certainty can be...
the foundation for a tower of truth.
What is the distance formula?
the square root of (x2-x1)2 + (y2-y1)2
Complete the life lesson: Carefully consider the outcomes of...
various scenarios
Complete the life lesson: Don't believe unsubstantiated claims, even if they sound scientific. Until...
you understand the issue for yourself be skeptical!
What is the largest prime number?
There is no largest. There are infinitely many primes.
What are some examples of golden rectangles in life?
-electrical socket -roof tile -whiteboard -door
An important property of golden rectangles is that every golden rectangle can be split into a square and a rectangle.
...
There seems to be no pattern in the prime numbers on a small scale. But the quantity of prime numbers fits into a function-even if individual prime numbers are unpredictable on a large scale.
...
What's the beginning of the Fibonacci sequence?
1,1,2,3,5,8,13,21,34,55...
What is the formula for the area of a triangle?
1/2bh
The Babylonian numbering system used a base of...
60. This is why we use 60 in a lot of our math today, like 60 seconds in a minute, 360 degrees in a circle)
Leonardo de Pisa
A 13th century Italian mathematician, he is credited for discovering Fibonacci numbers and the Fibonacci sequence, though the name comes from his father's name, not his own.
Who was Marin Mersenne?
A 17th century French mathematician, theologian, and musician, Mersenne compiled a list of all the Mersenne primes up to 2 to the 257th power minus 1. Unfortunately, his list had several mistakes in it. His Mersenne primes, as they would later be called, take the form of 2n-1.
Who was Pierre Fermat?
A French mathematician who wrote Fermat's Last Theorem in the margin of his copy of Diophantus' "Arithmetica," a famous ancient Greek math text. It stated that an + bn = cn has no solutions if n is greater than 2. This remained unsolved until 1994 when Andrew Wiles solved it using an area of math called elliptical curves.
What is the Riemann Hypothesis?
A famous unsolved problem about finding all the zeroes of the Riemann Zeta Function.
What is a conjecture?
A guess. It isn't certain but we think it's true.
Grigori Perleman
A man who in the early 2000s solve the Poincare conjecture and rejected the Fields medal and didn't give his mother healthcare.
Andrew Wiles
A mathematician who in 1994 solved Fermat's Last Theorem using an area of math called elliptical curves.
Who was Leonard Euler?
A nineteenth century German mathematician who built upon Euclid and Mersenne's ideas and proved that all even perfect numbers are connected to Mersenne primes.
Who was Karl Friedrich Gauss?
A nineteenth century German mathematician who discovered the Prime Number theorem in 1800. This theorem states that for very large values of n, the number of prime numbers less than n is approximately equal to n/ln(n). This wasn't proven until 1900.
Who was Al-Khwarizmi?
A ninth century Persian mathematician whose name gave us the word "algorithm." In addition, his book "Al-jabr wa'lmuqabalah" or "Restoring and Comparing" gave us the word "algebra."
What are rectangular numbers?
A number which can be represented by dots in a rectangle (i.e. 8, 44, 27)
What is a prime number?
A positive integer that is greater than 1 and is divisible only by itself and 1.
What is a golden rectangle?
A rectangle in which the ratio of the long side to the short side is the golden ratio.
Who were the Pythagoreans?
A religious/philosophical society in Ancient Greece. Plato was a famous follower. Founded by Pythagoras in the 6th century BC
What kind of numbering system do we use?
A representational numbering system.
What can you use to find the distance between points on a graph?
A right triangle and pythagorean theorem
What is a representational numbering system?
A system that uses different symbols to represent each number between 0 and 9. The Greeks also used this.
What is a positional numbering system?
A system where the position of numbers affect their value and where there is a base. For example, the Mayan numbering system has a base of 20.
Jacques Hadamard
Alongside de ville Poussin, he proved Gauss and Legendre's Prime Number Theorem in 1900.
Who was Euclid?
An ancient Greek mathematician and philosopher, Euclid knew about Mersenne primes (before Mersenne himself) and discovered that they were connected to perfect numbers.
What is a rational number?
Any decimal that 1) stops eventually 2) repeats numbers
What culture is credited with creating the modern numeral system?
Arabic mathematicians combined Greek and Indian ideas the create the modern numeral system.
What are prime triplets?
There's only one set prime triplets, 3, 5, 7, are three consecutive odd numbers, all of which are prime. But these are the only ones.
What are Pythagorean triples?
Three integers which satisfy the pythagorean theorem. Examples: (3,4,5) (6,8,10) (5,12,13) (12,16,20)`
Using proof by contradiction, prove that there are infinitely many prime numbers.
Claim: There are infinitely many prime numbers. Proof: Assume there is a largest prime number, m. Let's call all the numbers between 1 and m, N. N=(1x2x3x3x...xm)+1. Obviously, N is greater than m. By the Prime Factorization of Natural Numbers, there are only two possibilities for N: Either N is prime or N is a product of primes. If N is a prime, the proof if finished. But if N is a product of primes we still have a way to go. Since we don't know, we must proceed. If N is a product of primes, then we can choose one of those prime factors and call it Big-Prime. Now, let's pin down the value of Big-Prime. Does Big-Prime equal 2? If we divide 2 into N we see that there is a remainder of 1. Thus it cannot equal 2 because 2 doesn't divide evenly in N. Does Big-prime equal 3? If we divide 3 into N we see that there is a remainder of 1. Thus it cannot equal 3 because 3 doesn't divide evenly in N. So now we see that the remainder is always 1 when any number from 2 to m is divided into N. None of the numbers between 2 and m is a factor of N. Thus Big-Prime must be larger than m.
Using proof by contradiction, prove that there is no largest natural number.
Claim: There is no largest natural number. Proof: Assume there is a largest number (above statement is false), m. Fact: m+1 is a number greater than m which contradicts our assumption. Thus the claim is true.
Are square roots irrational?
Definitely. They are called "nice" or "algebraic" numbers because there is an equation to describe them.
What is the Prime Factorization Theorem (also called the Fundamental Theorem of Arithmetic)?
Every natural number can be written uniquely as a product of prime numbers.
Square numbers are part of a broader class of numbers. What are these numbers called?
Figurate numbers. These numbers result from the placing of points to create geometric figures.
What is the Fibonacci formula?
Fn+2=Fn + Fn+1
How would you go about checking for prime numbers?
For any number, n, we only need to check primes less than or equal to the square root of n.
What is the Prime Number Theorem?
For very large values of n, the number of prime numbers less than n is approximately equal to n/ln(n). Remember that (ln) is the inverse of the exponential function.
Who discovered the Prime Number Theorem?
Gauss and Legendre in 1800. It was proven by Hadam and Poussin in 1900.
This limit is called the...Mathematicians denote it using the symbol phi.
Golden ratio
What does the Prime Number Theorem attempt to answer?
How are prime numbers distributed?
What is the Collatz Conjecture?
If you pick a number and it's even you divide it by 2. If it's odd, you multiply by 3 and add 1. All the sequences interconnect. If you repeat this process, you always come back to 1, no matter where you started from. This is a CONJECTURE, so no one knows if this is true.
Where are Fibonacci sequences seen most vividly?
In nature, mostly plants and stuff like flowers, pinecones, and some vegetables.
Primes are...though it's hard to find big examples.
Infinite.
Arbitrarily large but finite vs. infinite
Infinity literally goes on forever. Even if the number is huge and seems to go on forever, it is actually finite.
Are there infinitely many twin primes?
It isn't clear.
Who discovered that all even perfect numbers are connected to Mersenne Primes?
Leonard Euler.
Who discovered Fibonnacci numbers?
Leonardo de Pisa in the 13th century.
Define inductive reasoning.
Looking at details and creating a general picture. -A reasoning process of generalizing from facts, instances, or examples. -Inference of a generalized conclusion from particular instances
Can rectangular numbers be prime?
NO.
Are any odd perfect numbers?
No one knows.
Are there infinitely many perfect numbers?
No one knows.
Is 1 considered a prime number?
No.
Can two consecutive numbers both be even?
No. Even=2n and Odd=2n+1.
Is 0.99 less than 1?
No. They are equal. The rule is that any two numbers have a real number between them. There isn't a real number in this case, as 0.99 is as close as you can get to 1, so they are the same.
What is the difference between number and numeral?
Number is a concept whereas numeral is the language or symbol used.
What are perfect numbers?
Numbers whose only divisors also add up to equal the number. For example, 6. The only natural numbers that divide evenly into 6 other than 6 itself are 1, 2, and 3. 3+2+1=6.
What are three examples of irrational numbers?
Pi=3.141... phi=1.618... e=2.718...
Who wrote Fermat's Last Theorem?
Pierre de Fermat wrote this theorem in the margin of his copy of Diophantus' "Arithmetica." This theorem remained unsolved until 1994, when Andrew Wiles solved it. It involved an area of math called Elliptical Curves.
What are Mersenne Primes?
Primes of the form 2n-1. We don't know how many there are. Not all of these numbers are prime.
Primes are seemingly...yet we know approximately how many primes can be found less than any large number.
Random.
Primes seem to become more and more...for larger numbers, yet we suspect the twin prime conjecture is likely true.
Rare.
What are natural numbers?
Represented by N, natural numbers are positive numbers, excluding 0.
What are whole numbers?
Represented by W, whole numbers are positive numbers, including 0.
What are integers?
Represented by a Z, integers are numbers along the number line, both negative and positive.
What is Proof by Contradiction?
Start by claiming the opposite of what you believe to be true. Once you disprove the impossible, whatever remains, however improbable it may seem is the truth.
What is a gnomon?
The piece that id added to a geometric shape to create the next shape in the sequence.
What does it mean to be written uniquely?
The prime numbers that you use and the order that you used them in CANNOT be changed.
Using the dependent Fibonacci pattern: 13/21=o.619 21/34=0.617 34/55=0.6181 55/89=0.618. What do you notice about this pattern. What is this pattern called?
The same number is produced by dividing a Fibonacci number with its previous term. This is called a limit. They're following the same pattern.
What is a direct proof?
Theorem: If A then B. Proof: Assume A is true and then reason inductively, therefore B.
True or false? There is no known formula for producing prime numbers.
True. An examples would be n2-n+40. This gives a prime number for all numbers less than 40, but it doesn't always work for larger numbers, such as 41. The pattern is nonexistent higher up.
Define Twin Primes.
Two consecutive odd numbers which are both prime. For example: 5 and 7, 11 and 13, 17 and 19, 29 and 31.
How do computers use prime numbers?
Used in encryption methods.
Why are assumptions important in math?
Without assumptions, it is impossible to draw conclusions. Assumptions are used all the time in proofs, both direct and by contradiction.
Can golden rectangles be split up into smaller golden rectangles?
Yes.
The modern numeral system is dependent upon...
Zero.
Complete the life lesson: Once you have an effective representation of this question...
a little experimentation will lead you to an answer.
What is the formula for pythagorean theorem?
a2 + b2= c2
What is Fermat's Last Theorem?
an + bn = cn has no solutions if n is greater than 2.
Complete the life lesson: Carefully understand and...
analyze the facts at hand
Complete the life lesson: Devising a good representation of a problem is frequently the...
biggest step toward finding a solution.
The Prime Number theorem says that as numbers get bigger and bigger, prime numbers become...
fewer and fewer.
Complete the life lesson: Experimentation is a powerful means...
for discovering patterns and developing insights
Complete the life lesson: Experimentation is an effective...
means of resolving difficult issues
Complete the life lesson: Often a clearer idea can be...
more potent than conventional wisdom
Fn denotes the...Fibonacci number.
nth
Complete the life lesson: Once you find and argument that resolves an issue, it is a great challenge to find a different argument. However, in attempting to find other arguments, we...
often gain further insight into and understanding of the situation. Also, the first argument we come up with may not be the best one.
Complete the life lesson: Look at problems from different...
perspectives.
In Fibonacci numbers, the pattern is depends on the
previous set.
