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I was going through the flashcards on Number properties. There was one question that kind of confused me. Ques: How many numbers are there between 3 and 6?

I answered 2 (4 and 5) because I thought that by 'numbers' it meant the counting numbers(digits) 1,2.3,4,.......

Can you please explain that why are decimals and fractions termed as numbers?

I was going through the flashcards on Number properties. There was one question that kind of confused me. Ques: How many numbers are there between 3 and 6?

I answered 2 (4 and 5) because I thought that by 'numbers' it meant the counting numbers(digits) 1,2.3,4,.......

Can you please explain that why are decimals and fractions termed as numbers?

The first thing I'll say is that we don't necessarily respond instantly here. If you have a particular question and you're a Magoosh customer, you can email help@magoosh.com. Perhaps you already did so.

When non-mathematical people hear the word "numbers," they typically think only of counting numbers {1, 2, 3, 4, ...}. When mathematicians think of numbers, at the very least, they think of everything on the number line. Mathematicians also think of exotic numbers off the number line (imaginary numbers, quaternions, etc.) that you don't need to know for the GMAT.

Two sets of numbers you need to know for the GMAT: counting numbers = {1, 2, 3, 4, . . . } integers = { . . . -3, -2, -1, 0, 1, 2, 3, . . . } Often the counting numbers are called "positive integers" on the GMAT.

There are other sets--you don't need to know the names, but you need to know that they are part of what mathematicians call "numbers" rational numbers = all fractions of the form (integer)/(integer), where of course the denominator is not zero (e.g. 1/2, 22/7, -3/5, etc) real numbers = every number on the number line, a continuous infinity of numbers; the real numbers include all the rational numbers and all the irrational numbers (e.g. \(\sqrt{3}, \pi, \dfrac{1+\sqrt{5}}{2}\), etc.)

The continuous infinity of real numbers between 0 and 1 is an infinity that's infinitely larger than the infinity you would get if you started counting 1, 2, 3, ... and never stopped. Continuous infinity is infinitely larger than countable infinity. All that is beyond what you need to know for the GMAT.

What you do need to know, quite rigorous, is that you can't be casual about the word "number." The word always implies all categories of numbers: positive, negative, and zero; integer, fractions, and decimals.

Does all this make sense? Mike
_________________

Mike McGarry Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

I was going through the flashcards on Number properties. There was one question that kind of confused me. Ques: How many numbers are there between 3 and 6?

I answered 2 (4 and 5) because I thought that by 'numbers' it meant the counting numbers(digits) 1,2.3,4,.......

Can you please explain that why are decimals and fractions termed as numbers?

The first thing I'll say is that we don't necessarily respond instantly here. If you have a particular question and you're a Magoosh customer, you can email help@magoosh.com. Perhaps you already did so.

When non-mathematical people hear the word "numbers," they typically think only of counting numbers {1, 2, 3, 4, ...}. When mathematicians think of numbers, at the very least, they think of everything on the number line. Mathematicians also think of exotic numbers off the number line (imaginary numbers, quaternions, etc.) that you don't need to know for the GMAT.

Two sets of numbers you need to know for the GMAT: counting numbers = {1, 2, 3, 4, . . . } integers = { . . . -3, -2, -1, 0, 1, 2, 3, . . . } Often the counting numbers are called "positive integers" on the GMAT.

There are other sets--you don't need to know the names, but you need to know that they are part of what mathematicians call "numbers" rational numbers = all fractions of the form (integer)/(integer), where of course the denominator is not zero (e.g. 1/2, 22/7, -3/5, etc) real numbers = every number on the number line, a continuous infinity of numbers; the real numbers include all the rational numbers and all the irrational numbers (e.g. \(\sqrt{3}, \pi, \dfrac{1+\sqrt{5}}{2}\), etc.)

The continuous infinity of real numbers between 0 and 1 is an infinity that's infinitely larger than the infinity you would get if you started counting 1, 2, 3, ... and never stopped. Continuous infinity is infinitely larger than countable infinity. All that is beyond what you need to know for the GMAT.

What you do need to know, quite rigorous, is that you can't be casual about the word "number." The word always implies all categories of numbers: positive, negative, and zero; integer, fractions, and decimals.

Does all this make sense? Mike

Yes!! Thanks for your reply.

Also may I ask (kind of a basic question), the difference between counting nos. and whole nos. is zero (0). Counting nos. do not include zero but we can say that there are zero files in this box, then why do we keep it out of natural nos. ?

Why the list is called whole numbers when we include zero in it ?

I know they are kind of very basic questions and I don't need to go in that much depth but this is kind of interesting as well as confusing and it would be great if you could clear this confusion.

Also may I ask (kind of a basic question), the difference between counting nos. and whole nos. is zero (0). Counting nos. do not include zero but we can say that there are zero files in this box, then why do we keep it out of natural nos. ?

Why the list is called whole numbers when we include zero in it ?

I know they are kind of very basic questions and I don't need to go in that much depth but this is kind of interesting as well as confusing and it would be great if you could clear this confusion.

Great questions! First of all, as I am sure you understand, these questions and the answers I will give have ZERO to do with the GMAT. While these are wonderful things to explore, they absolutely will not appear on the GMAT.

The term "counting numbers" is a more colloquial, non-technical name. The official mathematical name is the natural numbers. These are the numbers found in Nature. I can show you two trees, or seven birds, etc., but I can't show you zero of anything. Zero of something is a human imagination: it's not something we actually find in our sensory experience of the natural world. It's funny. In a poem entitled "Snow Man," the poet Wallace Stevens talks about someone who "beholds nothing that is not there and the nothing that is." Beholding "nothing" is a poetic idea.

Even with the idea of counting---I can count to 1 or 2 or any counting number, but can I really "count" to zero? Does "counting to zero" really mean the same thing as ordinary counting? Is apprehending a presence the same as conceptualizing an absence?

Notice that we are getting into some Zen territory here! The Baso said: "If you have a staff, I will give it to you. If you have no staff, I will take it away from you." We could debate for days about the existence or non-existence of zero or nothingness!

Now, as for the term "whole numbers," a set that includes zero--I believe it means "whole" as in "undivided." Each counting number is a whole, undivided thing. There are not fractions or pieces or parts involved when we have a counting number. Much in the same way, when we are dealing with the number zero, there are no extra pieces or fractional parts lying around. In that sense, there is a undivided wholeness to zero as exists for natural numbers.

Of course, we are on tricky ground with that thought, because if zero is undivided, why isn't -3? We get around this by using a Latin word. The word "integer" comes from the Latin word for "wholeness"--from this root, we also get the English word "integrity," a word for a person whose words & actions are aligned with his wholeness. It's unclear whether these two names, "whole numbers" and "integers" are 100% justified, but the sets are meaningfully different, and we need some words for them. Language is funny: it's never 100% logical.

Historically, the ancient Greeks were totally aware that positive fractions and even irrational numbers were numbers. It was the Pythagorean school that proved that \(\sqrt{2}\) was irrational. It took centuries in the middle ages for zero to be accepted as a number, and after that, it took a while for negative numbers to be fully accepted. By the time of Rene Descartes, the man who designed the x-y plane, the mathematical community was pretty much comfortable with all the real numbers. Then, it took mathematicians longer to accept the idea of imaginary numbers and complex numbers, but even those are 100% accepted as numbers now, along with some far more exotic beasts.

All fascinating stuff, and absolute none of this do you need to know for the GMAT!

Does all this make sense? Mike
_________________

Mike McGarry Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)