Odd and even numbers

[user789653241]

Ds(9, yr5) asked me if decimal numbers can be defined as even or odd. I said no, but haven't got a clue why.

Can anyone give us some explanations please?

Essentially the answer is because when you divide it by two it doesn’t give you a whole number as an answer.

Zero is even

https://en.m.wikipedia.org/wiki/Parityoff_zero

As far as primary school maths goes, all it is is that even by definition means "can be divided by 2 leaving a whole number (integer)". Not "can be divided by 2 leaving any rational/real number".

In grown up maths, divisibility is a sensible concept as long as you have what's called an integral domain (broadly a structure with +,-, 0 and that behave as we'd expect them to). A field (broadly a structure with +,0,- , ,1 and /) is an example of an integral domain so you can define divisibility in it, but as you can divide numbers in a field, everything is divisible by everything.

Integers are an example of an "integral domain"; rational numbers are a "field".

I

What could also be interesting, is why we have/need the concept of odd and even in itself. We don't for example have words for 'threesomeness' where a number divisible by three is called threeit, one that leave a remainder 1 is called three-be and one that leaves a remainder two is called three-less (or whatever).

So why is the concept of odd and even so useful it has a name?

ps. Does your DS know that zero was a great invention? They didn't used to have it because you couldn't have something that represented nothing.

I don't get why decimals can't be even or odd.

2.4 is surely even as it can become 1.4...

He sounds like A deep thinker!
My dd asked similar questions at that age, and is now in year 9 being accelerated to do her maths GCSE next summer. She so did not get this from me.
So watch out op, he may be a rapid progressor.

Annor I don't get why decimals can't be even or odd.

a) because the definition is "Any integer that can be divided exactly by 2 is an even number." The word integer means whole number

b) because if you let any decimal be odd or even it becomes meaningless. Any rational number can be divided into two. Any decimal can be written with an extra 0 on the end, thus turning it into an 'even' number. So 1.1 can be written 1.10.

You can't pick or choose which bit of a definition you want to use.

Just like the definition of a square goes something like:
A 2 dimensional closed polygon with 4 straight sides of equal length and each of the 4 angles being a right angle.

If you drop off the bit about equal length you could have a rectangle, and if you drop off the bit about right angles you could have a rhombus.

Only integers can be classed as odd or even numbers.

Integers are whole numbers.

Teen

Never learned that definition at school. Only whether it could be divided evenly or not. I admit maths is not my strong point at all. I don't even see how 1.1 is the same as 1.10 😂 Bowing out.

Teen, we were talking about 0 recently. They are doing roman numerals and that they didn't have concept of 0.

I was googling for this decimal can't be even or odd as well as ask here.
Most of the answers were just stating the fact, but some had given more complicated answer. Since I am not mathy myself, it was still incomprehensible to me, but one made a sense.
It said, since rational decimal numbers can add 0s after the last digit, that concept of odd/even won't apply.
So, if my understanding is correct, you can even add 0 to 3 or 5, to make it 3.0 or 5.0, or add 0 to 3.5 and make it 3.50, etc., which definitely doesn't follow the rule, since 3 is odd, but dding 0 to make it 3.0 doesn't make it even. Is this correct way to think?

Ummm. Don't think I like the wording of 'adding 0' myself.

Though that does prompt the question. Could 4.0 (which is equivalent to 4) be considered an integer? I think not as it hasn't been written as one. I am going to tag two of the other MN mathematicians @noblegiraffe and @DadDadDad to see whether they would like to join in the discussion.

Zero helps other digits keep their place value, but also has the concept of nothingness. So 1.2 means 1 whole and 2 tenths, and 1.20 means 1 whole, two tenths and zero hundredths. But the zero in this case is superfluous, just as we don't write 86 as 086 where the zero means no hundreds.

An odd number plus an odd number is even 3 + 1 = 4
An odd number plus an even number is odd 3 + 2 = 5
An even number plus an even number is even 2 + 2 = 4

Now look at zero

3 + 0 = 3 which is odd
2 + 0 = 2 which is even

So 0 cannot be odd or the rules of addition wouldn't work.

Also, every even number is sandwiched bewteen two odd numbers

2 is between the odd numbers 1 and 3
4 is between the odd numbers 3 and 5
-2 is between the odd numbers -3 and -1

0 is between the odd numbers -1 and 1

An even number when divided by 2 gives another whole number

8 divided by 2 is 4
4 divided by 2 is 2
2 divided by 2 is 1
0 divided by 2 is 0

Odd numbers give a fraction when divided by 2

3 divided by 2 is 3/2 or 1.5
1 divided by 2 is 1/2 or 0.5
0 divided by 2 is 0 which is not a fraction

Could 4.0 (which is equivalent to 4) be considered an integer?

As long as you are happy to regard the integers as a subset of the real numbers then yes I would say 4.0 = 4 belongs to that subset so is an integer. If you want to really confuse people, 3.99999.... (3.9 recurring) is also equal to 4, so is also an integer

Just to show that cat is correct:

y = 3.99999....
10y = 39.9999.....

10y-y=36
9y=36
y=4

so 3.999.... = 4 . QED.

In computer programming, 3.0 is not an integer.

3.0 implies some sort of definition of precision so I'd argue it is not an integer.

Never learned that definition at school. Only whether it could be divided evenly or not. I admit maths is not my strong point at all. I don't even see how 1.1 is the same as 1.10 😂 Bowing out.

Briefly,

Every number can be divided evenly. It’s whether the result is a whole number or not.
Half of 7 is 3.5. But that doesn’t make 7 an even number. Half of 8 is 4 so that is.

And your second question, 1.1 is the same as 1.10 in the same way that 1 can be written as 01
It’s just adding zeros in the place where there is no need, because we already assume that column has a zero value.

I think we have to assume we're talking about maths and number here given the context OhYou.

You could equally store 3 or 3.0 in a text format data field if you want to talk about data types. Or I could define a function "IsAnInteger" applied to my decimal data type which would return 1 if applied to 3.0 and 0 if applied to 3.1.

If we're talking about precision, "3.0 to 1 decimal place" represents the interval [2.95, 3.05). But then you'd better not try to do arithmetic with it, or you will have problems with accumulation of rounding errors.

I'd also argue that 3.0 is not an integer because it could actually be 3.01 expressed to one decimal place, so not a whole number.

Concept of dividing goes something like this:

• you can't divide 7 by 2, it doesn't go
• 7 divided by 2 is 3 remainder 1
• 7 divided by 2 is 3.5
• 7 divided by 2 is 3.5, but sometimes you'll need to round up or down the answer if you are dealing with real life (eg 7 children and some tandem bicycles, how many do you need? or 7 socks, how many whole pairs can you make)

Sorry, OP. We have gone off on a straight line just touching a circle.

It's probably one of those problems that sounds simple on the surface but there are people out there who will have written undergraduate dissertations or spent years researching for a PhD on the subject.

Hmm, 3.9 recurring is equal to 4 and 4 is an integer but I'd be reluctant to classify 3.9 recurring as an integer. It feels like saying that 0.2 is equal to 1/5 and 1/5 is a fraction so 0.2 is a fraction.

What I mean there is that a number can have a particular property or label that doesn't transfer across to all other representations of the same number.