An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.
The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold
(
Z
)
{\displaystyle (\mathbb {Z} )}
letter "Z"—standing originally for the German word Zahlen ("numbers").ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.
I have 2 questions.
1)what is the remainder with 100! is divided by 103? explain your answer
2)a = 238000 = 2^4 x 5^3 x 7 x 17 and b=299880 = 2^3 x 3^2 x 5 7^2 17. Is there an integer so that a divides b^n? if so what is the smallest possibility for n?
the first one i have no...
Hello all,
I've been asked for a graduate level course to do a proof using induction that shows that nCr always turns out to be an integer. I thought that I might use Pascal's triangle somehow and the fact that nCr is equal to n! / r!(n-r)! (I saw a brief explanation of this while doing a web...
Hai
COuld you please help the formula as I am not able to identify the question below:
Question:
For a Traingle with a perimeter of 30cm, what is the integer value of the longest possible side ?
REgards
aprao
Can anybody explain what appear to be discrepancies in the way the following expressions are interpreted by Hugs (Haskell98 mode) ?
Main> div -6 4
ERROR - Cannot infer instance
*** Instance : Num (b -> a -> a -> a)
*** Expression : div - fromInt 6 4
Main> div (-6) 4
-2
Main> -6...
I just completed Trigonometry and College Algebra, and I'm heading into Calculus, so i thought i would get a head start on the material. So right now I'm working out of an old calculus book i got at the library. Then i came across this problem:
A dial-direct long distance call between two...
for n- fixed integer prove that
phi(x)=n has a finite number of solutions
I looked at 2 cases when x is even and when x is odd
1) if x is even then phi(2x)>phi(x) and I showed why it has a finite number of solutions
2) I'm not sure how to show for the case when x is odd.. any ideas?
thanks :)
The terms of this series are reciprocals of positive integers whose only prime factors are 2s and 3s:
1+1/2+1/3+1/4+1/6+1/8+1/9+1/12+...
Show that this series converges and find its sum.
this is my first time writing here. i hope someone can help me with this question.
I'm not a very logical person, and I would hardly consider math a strength so, I'm stuck with these proofs:
1. Prove that every positive integer, ending in 5 creates a number that when squared, ends in 25
2. Prove that if n is an even positive integer, then n^3 - 4n is always divisible by...
The Q is: show that the number of partitions of n within Z+ where no summand is divisible by 4 equals the number of partitions of n where no even summand is repeated
Here is what I got so far
Let the partition where no summand is divisible by 4 be P1(x)
Let the partition where no even...
Seem to remember reading about RF experimenting with non-integer differentiation. I found it quite interesting to play with.
I started with 'half' differentiation.
e.g. f(x) = x^2 , D[1]f = 2x , D[2]f = 2
but what about non-integer diff?
e.g. D[0.5]f = px^1.5
what is p?
Clearly...
I really need someone to answer this by tomorrow 12/01 10:50 am pacific time...
1) Derive the following, for any integer k.
infinity
SUM OF: 1/(n^(2k)) = ((2(pi))^(2k)(-1)^(k+1)B2k ) /(2(2k)!)
n=1
where Bn is defined by the following, for |x| < 2(pi).
````````````infinity
(x)/(e^(x)-1)...
if
a^k =1
and
a \in \mathbb{C}
k \in Z^+
and for some k
A = \{a|a^k = 1\}
does
|A| = k
?
edited: becasue the real numbers are a subset of the complex numbers