Need to know symbolic representations of odd and even numbers. Its closely related to Number theory.What is symbolic representations of odd and even numbers?
If you want symbols for the sets of all odd or even numbers, you can use 2N+1 and 2N, where N is the set of natural numbers.What is symbolic representations of odd and even numbers?
Evens: {x: x = 2n for all n [an element of] N},
Odds: {x: x = 2n-1 for all n [an element of] N}
The above is written on the basis that N, the set of natural numbers consists of {1, 2, 3, ...}.
Some conventions define N as {0, 1, 2, ...}. With this definition, the set of all odd positive integers would be:
{x: x = 2n+1 for all n [an element of] N}
If you are looking for a symbol that represents the set of all odd numbers or even numbers similar to R = Reals, Q = rationals and N = natural numbers, there does not appear to any in common usage.
This task offers the opportunity for students around CSF level 5 or early CSF level 6 to use symbolic representations of even
and odd numbers, together with various structural principles of algebra, to develop informal proofs, by deduction, of
generalisations developed in Task 3. These principles include the expansion of brackets and grouping like terms.
Other conjectures are offered below as opportunities for students to develop informal proofs. While students will probably need
to specialise these generalisations (and give instances of them in their abstract symbolic form) to interpret them, the intention is
that they develop their ability to develop and express principled arguments.
5.1 Proving the generalisations developed
in Task 3, for example:
even + even = even
odd + odd = even
odd + even = even
even + odd = even.
Also, conjectures about the products:
even 脳 even
odd 脳 odd
odd 脳 even, and the differences:
even 鈥?even
odd 鈥?odd
odd 鈥?even
even 鈥?odd.
Also:
Are the following statements always true
(a and b are positive integers or zero):
If a + b is even then so is a 鈥?b?
Is the converse of this statement true?
If a + b is even and a 鈥?b is even then ab
is even?
The product of two consecutive whole
numbers is even?
(Also see the extensions for this Task over
the page).
Use of abstractions, algebraic
principles and instantiations (as
needed) as tools to develop
principled arguments using the
processes of Specialising, Correcting,
Generalising, Explaining, Justifying,
and Convincing.
For example:
Prove the statement that the sum of
an odd and even number always
gives an odd number.
Let the odd number be 2n + 1
(n is zero or a positive integer).
Let the even number be 2m (m
is zero or a positive integer).
So, the sum of the odd and even
number may be represented as:
2n + 1 + 2m.
This can be restated as:
2n + 2m + 1, and again as:
2(n + m) + 1.
Now, 2 (n + m) must be even
(we have shown earlier that
doubling any number yields an
even number).
So, 鈥?(n + m) + 1鈥?represents an
even number plus one. We have
shown earlier that one more than
an even number is always odd.
Thus,
2(n + m) + 1 is odd, and the
statement that 鈥榯he sum of an
odd and even number always
gives an odd number鈥?is always
true.
Reasoning and strategies
MARSR 501, 502 through to 601,
602 and 603
MARSS 501, 502, 503, 504
through to 601, 602 and 603, 604
Number
As for Task 4
Algebra
MAALE 501, 502, 503 through to
601, 602, 603, 604 and 605, 606
Engagement Links to outcomes in the
Mathematics CSF and
Resources
Interpretations about the
ways of mathematical
thinking involved
You should open these two links you will get a lot about your answer
http://www.vcaa.vic.edu.au/prep10/csf/pu鈥?/a>
http://resources.mhs.vic.edu.au/fs1sl/CS鈥?/a>
2n for evens, 2n+1 for odds. not satisfied?
i use them a lot :)
