Identifying even and odd numbers
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- Even/Odd Number Maze!
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For example, numbers such as 11, 23, 35, 47 etc. All the numbers ending with 0,2,4,6 and 8 are even numbers. For example, numbers such as 14, 26, 32, 40 and 88 are even numbers. If we divide a number into two groups with an equal number of elements in each, then the number is an even number.
In the case of odd numbers, we get a remainder of 1 while grouping. In the case of a remainder, a number is an odd number. The elementary skills in number sense are useful in later grades for learning math, science, and communication systems. We apply the concept in designing circuits using logic gates and binary codes. In ancient mathematics, the learning of geometric shapes started from categorizing the shapes based on the number of sides as even and odd. Before the 20th century, definitions of primality were inconsistent, and significant mathematicians such as Goldbach , Lambert , Legendre , Cayley , and Kronecker wrote that 1 was prime.
This definition can be rationalized by observing that it more naturally suits mathematical theorems that concern the primes.
For example, the fundamental theorem of arithmetic is easier to state when 1 is not considered prime. It would be possible to similarly redefine the term "even" in a way that no longer includes zero. However, in this case, the new definition would make it more difficult to state theorems concerning the even numbers. Already the effect can be seen in the algebraic rules governing even and odd numbers. Inserting appropriate values into the left sides of these rules, one can produce 0 on the right sides:.
The above rules would therefore be incorrect if zero were not even. For example, one test study guide asserts that even numbers are characterized as integer multiples of two, but zero is "neither even nor odd". Making an exception for zero in the definition of evenness forces one to make such exceptions in the rules for even numbers.
From another perspective, taking the rules obeyed by positive even numbers and requiring that they continue to hold for integers forces the usual definition and the evenness of zero. Countless results in number theory invoke the fundamental theorem of arithmetic and the algebraic properties of even numbers, so the above choices have far-reaching consequences. For example, the fact that positive numbers have unique factorizations means that one can determine whether a number has an even or odd number of distinct prime factors.
Since 1 is not prime, nor does it have prime factors, it is a product of 0 distinct primes; since 0 is an even number, 1 has an even number of distinct prime factors. This apparently trivial observation can provide a convenient and revealing proof explaining why an odd number is nonzero.
A classic result of graph theory states that a graph of odd order having an odd number of vertices always has at least one vertex of even degree. The statement itself requires zero to be even: the empty graph has an even order, and an isolated vertex has an even degree. The appearance of this odd number is explained by a still more general result, known as the handshaking lemma : any graph has an even number of vertices of odd degree. Sperner's lemma is a more advanced application of the same strategy. The lemma states that a certain kind of coloring on a triangulation of a simplex has a subsimplex that contains every color.
Even and Odd Numbers
Rather than directly construct such a subsimplex, it is more convenient to prove that there exists an odd number of such subsimplices through an induction argument. The fact that zero is even, together with the fact that even and odd numbers alternate, is enough to determine the parity of every other natural number.
This idea can be formalized into a recursive definition of the set of even natural numbers:. This definition has the conceptual advantage of relying only on the minimal foundations of the natural numbers: the existence of 0 and of successors. As such, it is useful for computer logic systems such as LF and the Isabelle theorem prover. Indeed, "zero is an even number" may be interpreted as one of the Peano axioms , of which the even natural numbers are a model. The classic point in polygon test from computational geometry applies the above ideas. To determine if a point lies within a polygon , one casts a ray from infinity to the point and counts the number of times the ray crosses the edge of polygon.
The crossing number is even if and only if the point is outside the polygon.
Odd Man Out: A Lesson in Even & Odd Numbers – Teacher-Created Lesson Plan | Common Sense Education
This algorithm works because if the ray never crosses the polygon, then its crossing number is zero, which is even, and the point is outside. Every time the ray does cross the polygon, the crossing number alternates between even and odd, and the point at its tip alternates between outside and inside. In graph theory, a bipartite graph is a graph whose vertices are split into two colors , such that neighboring vertices have different colors.
If a connected graph has no odd cycles , then a bipartition can be constructed by choosing a base vertex v and coloring every vertex black or white, depending on whether its distance from v is even or odd. Since the distance between v and itself is 0, and 0 is even, the base vertex is colored differently from its neighbors, which lie at a distance of 1.
In abstract algebra , the even integers form various algebraic structures that require the inclusion of zero. The fact that the additive identity zero is even, together with the evenness of sums and additive inverses of even numbers and the associativity of addition, means that the even integers form a group.
Even & Odd Numbers
Moreover, the group of even integers under addition is a subgroup of the group of all integers; this is an elementary example of the subgroup concept. Since the even integers form a subgroup of the integers, they partition the integers into cosets. Analogously, the alternating group is a subgroup of index 2 in the symmetric group on n letters. The elements of the alternating group, called even permutations , are the products of even numbers of transpositions.
The identity map , an empty product of no transpositions, is an even permutation since zero is even; it is the identity element of the group. This formulation is useful for investigating integer zeroes of polynomials. There is a sense in which some multiples of 2 are "more even" than others. Multiples of 4 are called doubly even , since they can be divided by 2 twice.
Not only is zero divisible by 4, zero has the unique property of being divisible by every power of 2 , so it surpasses all other numbers in "evenness". One consequence of this fact appears in the bit-reversed ordering of integer data types used by some computer algorithms, such as the Cooley—Tukey fast Fourier transform.
This ordering has the property that the farther to the left the first 1 occurs in a number's binary expansion , or the more times it is divisible by 2, the sooner it appears. Zero's bit reversal is still zero; it can be divided by 2 any number of times, and its binary expansion does not contain any 1s, so it always comes first. Although 0 is divisible by 2 more times than any other number, it is not straightforward to quantify exactly how many times that is.
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For any nonzero integer n , one may define the 2-adic order of n to be the number of times n is divisible by 2. This description does not work for 0; no matter how many times it is divided by 2, it can always be divided by 2 again. Rather, the usual convention is to set the 2-order of 0 to be infinity as a special case. The powers of two—1, 2, 4, 8, In the 2-adic numbers , such sequences actually converge to zero.
The subject of the parity of zero is often treated within the first two or three years of primary education , as the concept of even and odd numbers is introduced and developed. The chart on the right  depicts children's beliefs about the parity of zero, as they progress from Year 1 to Year 6 of the English education system. The data is from Len Frobisher, who conducted a pair of surveys of English schoolchildren. Frobisher was interested in how knowledge of single-digit parity translates to knowledge of multiple-digit parity, and zero figures prominently in the results.
In interviews, Frobisher elicited the students' reasoning. One fifth-year decided that 0 was even because it was found on the 2 times table. A couple of fourth-years realized that zero can be split into equal parts. Another fourth-year reasoned "1 is odd and if I go down it's even.