Prove that (f+g)(x) is an odd function, if f and g are odd functions (Stewart, Calculus)

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Suppose f(x) and g(x) are odd functions. Prove that (f+g)(x) is also an odd function.  Answer:  1. Strategy By definition, f is an odd function if and only if f(-x) = - f(x) To show (f+g) is an odd function, we need to show (f+g)(-x) = - (f+g)(x) 2. Explanation Since $f(x)$ and $g(x)$ are odd functions $\Rightarrow f(-x) =-f(x)$ and $g(-x) =-g(x)$ By definition of sum of functions. $(f+g)(-x) =f(-x)+g(-x)$ $=-f(x)-g(x)$ $=-(f(x)+g(x))$ $=-(f+g)(x)$ (by definition of sum of functions) $\Rightarrow(f+g)(-x) =-(f+g)(x)$ $\Rightarrow f+g$ is an odd function. Q.E.D. 
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[Math] Taking a square root of a number: finding an upper limit for factors of a non-prime number (Eratosthenes)

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Eratosthenes has discovered that "a non-prime number has its factor (other than 1) that is less than the square root of the number." 


1) Task: finding an upper limit for factors of a non-prime number. 

 

2) Motivation: suppose number = 100

square root (100) = 10

factors of 100 = {1, 2, 4,5, 10, 20, 25, 50, 100} 

Note that factors of 100 exist before the square root of 100 (= 10). Namely, 1, 2, 4, 5 are those. With exception of 1, we have found prime factors of 10 that are less than 2,4, and 5. That's good enough information to determine that 100 is not a prime number.

 

3) An idea for mathematical proof would be:

1. suppose a number is a non-prime number. That is, this number = M * N where M>N > 0 

2. multiplying N to both sides of the inequality  M>N results M*N > N^2 

3. by design, this number = M*N > N^2

4. taking the square root, we have (M*N)^(1/2) > N 

conclusion: a factor of a non-prime number is less than this number itself. 

 

4) Tests

Test 1: let number  = 100 = 5*20  = 2*50 = 4*25 = 1*100 = 10*10 

square(100) = 10 -> there exists a factor(other than 1) of 100 that is less than 10: namely, 2,4,5.

 

Test 2: let number = 56 = 1*56 = 2*28 = 4*14 = 7*8 

square(56) = 7.48331477355....  -> there exists a factor(other than 1) of 56 that is less than 7.48: 2,4,7 

 

End

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