How do I disprove the statement that there exists a real number x such that x^6+x^4+1=2x^2?
Okay, let's show x^6+x^4+1 > 2x^2 Generally in these types of questions you are supposed to use calculus, because calculus is always a 100% guaranteed way of determining the shape of the graph, even if it gives you messy numbers. Let's find the minimum value of f(x) = x^6 + x^4 - 2x^2 + 1 By differentiating: f'(x) = 6x^5 + 4x^3 - 4x = 2x(3x^4 + 2x^2 - 2) And again: f''(x) = 30x^4 + 12x^2 - 4 What are the roots of f'(x)? Well one root is x = 0, and by plugging it into f''(x) we get f''(x) to be -4, hence the graph reaches a local maximum at x = 0 The other roots are solutions to 3x^4 + 2x^2 - 2 = 0 The quadratic equation tells us the roots correspond to: x^2 = (root(7) - 1)/3 Okay, we hope the graph reaches its minimum at these points. Let's check by plugging into f''(x): f''(x) = 30x^4 + 12x^2 - 4 = 10(3x^4 + 2x^2 - 2) - 8x^2 + 16 = 16 - 8x^2 root(7) is less than 3, hence (root(7) - 1)/3 is less than 2/3, therefore f''(x) is positive, hence the graph reaches it's minimum at these two points! So to prove the answer, we just have to show that the value of f at these minimum points is positive. The graph must be bigger everywhere else, because the graph tends to +infinity as x tends to +/- infinity. f(x) = x^6 + x^4 - 2x^2 + 1 = (x^2)/3 * (3x^4 + 2x^2 - 2) + (x^4)/3 - (4x^2)/3 + 1 = 1/9 * (3x^4 + 2x^2 - 2) - 14(x^2)/9 + 11/9 = 1/9 * (11 - 14x^2) and we know x^2 < 2/3, so this is bigger than: 1/9 * (11 - 28/3) > 0 So this minimum value is bigger than 0, hence the graph is always bigger than zero. QED
Can I say that every real number is a complex number but every complex number is not a real number?
Yes. Any real number is a special complex of the form. a+b.i where b=0.On the other hand, complex numbers cannot be seen as real numbers, as complex numbers do not form an ordered field ( that is, a field where the ordering is compatible with the basic operations of sum and product).
Why am I being taught complex numbers that have no use in my future rather than some useful stuff?
To start with, Randall Monroe had something interesting to say about this: xkcd: Forgot Algebra (he often does). As he said, the fun parts of life are optional.I have my own thoughts about this as well. First of all, you don't know what you are going to need in life. I had no idea that I was going to be a mathematician until I was partway through college---I thought I was going to be a novelist. If you were to insist on removing everything "useless" from your curriculum, you are effectively closing the door on becoming a physicist, or mathematician, or engineer; at the very least, you would make it far, far harder for yourself to pursue such a profession. Part of the goal of a public education is to give you a wide, common set of knowledge to prepare you for various contingencies.The other part of it is that public education is not there primarily to prepare you to be a productive professional; it is there to prepare you to be a citizen. We need a wide, common set of knowledge for all of us to even stand a chance of being able to communicate to each other.This is a huge problem for scientists and mathematicians. Often, we are working in areas that require a vast amount of specialized knowledge, and yet we still need to communicate with ordinary people (if only to explain why they should continue funding our work). It is difficult enough to do that as it is. (There is Quora page about this, actually: How do mathematicians working on highly abstract topics explain what they do to non-math, non-science people?)I agree that mathematics education is need of serious reform in this country---algebra and proofs should come earlier, experimentation should be encouraged, more context and appreciation for the big ideas should be provided, etc., etc. However, cutting out all of the "useless" bits is a bad approach to this problem.EDIT: I wrote "in this country" without thinking: in my case, that is the US. Of course, mathematics education could use reform in many other countries as well.
Plz prove that 2 is the only even prime number.?
Definition: A number p > 0 is prime if and only if the only divisors of p are 1 and p. Definition: A number x is even if and only if there exists an integer k such that x = 2k. To prove that 2 is the only even prime number we need to prove two things: 1) 2 is an even prime number 2) If there existed another even prime number, then it must be 2 Proof of 1): 2 is even since 2 = 2 * 1 2 is prime because it has divisors 1 and 2 Proof of 2): Assume there existed another even prime number say m =/= 2. Since m is even we see that m = 2k for some integer k. Thus, 2 is a divisor of m. Since m is prime, we know that 1 and m are the divisors of m. Thus, since m =/= 2 we have three distinct divisors of m (which are 1, 2, m). This contradicts the definition of prime unless m = 2. Therefore, 2 is the only even prime number.
Why do irrational numbers exist (why is the square root of 2 irrational, and not rational)?
Why do such numbers exist ? Interesting question ; I am not an expert on this meta thinking , but I will give my thoughts.Let us consider a fictional story of a prehistoric intelligent person who start by knowing only 0 (representing nothing) and 1 (representing something which exists)He invented numbers like 2,3,4,5 for his benefit, of counting his property of goats or cows. He then invents addition. Then inverse/reversal gives subtraction, and negative numbers.He might ask why do number like -2,-3,-4 exist, because he currently has no use for these. But when he thinks of lending and borrowing goats, he will use negative numbers.He might use N store houses, each of which has M bags of grain. He will invent multiplication to count his bags as N*M. Then inverse/reversal gives division. He might wonder why 1/2,3/4,5/6 exist, but when he thinks of dividing his property to his kids, he will think of these fractions.Because of multiplication, he might think of squares. The inverse/reversal gives square roots. He might then ask why root 2 exists , because he can never divide his property into sqrt(2) pieces. But when he wants a square of certain area, root 2 will be used in his calculations.He might think of a cubic store house, and reversal gives cube roots.At each point, he is inventing new operations, and inverse operations, which give new numbers, which are not available with his previous numbers or operations. He is constantly generalising, which is a basic human trait, and very good for mathematics.He now draws the number line and finds even more numbers like pi or e, which are algebraic, or computable or transcendental.Now he goes into two dimensions, and starts considering complex numbers.So conclusion (or moral of the story) is that given some operations, we can define some numbers, and many numbers might not be definable by those operations. Using addition,subtraction, multiplication, and division, we can define rational numbers, but we can not define some numbers which we call irrational numbers. There is nothing special about root2 in this view. Even if root2 happened to be rational, there would have been some other irrational number.Maybe now our fictional character will ask why does infinity exist. He will take this as something beyond comprehension or GOD or everything. He will find different type of infinities.
Which of these statements about sums of numbers is true?
The first is false (example 5+7) and the second is true: Odd prime x odd prime = odd number. Even prime x even prime = even number. Odd number + even number = odd number.