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# Real Numbers

Subsets of the Real Numbers

by

Tweet## Garry Carpenter

on 24 August 2010#### Transcript of Real Numbers

Or, if you rather think in terms of decimal numbers, these are all the decimal numbers that either STOP somewhere like

5

OR

2.97

OR

14567.3900723458679490202981834 Integers Rational Numbers 34343434343434 343434...FOREVER Whole Numbers Another way to say "fraction" is ratio. Hence the name RATIOnal numbers. Let's say this box represents all the real numbers. One special subset of rational numbers it the set of can never be the same as The Real numbers can be broken into two disjoint sets. Irrational Numbers do not include any 343434343434 Disjoint means non-overlapping. So something in one set cannot ever be part of the other. There is no such thing as a number that is BOTH rational and irrational, the sets are disjoint. which are not the same as All of the Real numbers that CANNOT be written as a fraction of two integers. A subset of the Integers is the set of Natural Numbers are disjoint from Take just the ZERO out of the Whole Numbers and you are left with the set of May not seem like much, but this turns out to be a really big deal. Real Numbers Rational Numbers Real numbers have two basic properties What are they? What is a FIELD???? Most cows are truly outstanding in their fields! Most cows are truly outstanding in their fields! Most cows are truly outstanding in their fields! They are an ORDERED field. They have a Least Upper Bound property. MEANS: Every number in the set can

be put in order from smaller to larger. Most cows are truly outstanding in their own field. No, not that kind of field.

Look it up! What is a mathematical

field? Report what you find. HUGE! Let’s say we have a set from A to B on the real number line. This set contains all numbers between A and B (doesn’t matter if A and B are in the set or not.) See all these points that are above B?

These are upper bounds for the set. This property just guarantees that there is one of these upper bounds somewhere that is smaller than all the rest… the Least of the Upper Bounds. Huh?!? (What kinds of Real Numbers are there?) Classification. Then what is out here? Any real number that can be written as a fraction of integers. Taken together, these two properties tell us that

the real numbers can be put into a single ordered line

and that the line has no holes in it anywhere. These

are critical features of the... 5.222222222222222222222.....

and

322.773216543654365436543..... OR decimal numbers that go on forever but at some point start repeating themselves. Like... In the box representing Rational Numbers there are some basic subsets. Integers are rational numbers with a denominator of 1

(so they can be written without using a fraction) -5 -123 -1 -34 -1000 0 1 2 3 4 5... 120 -8 zero 18 12,000,003 47 24 246 These can be described as the NON-NEGATIVE Integers. Why can't we make it simple and call them the POSITIVE Integers? Because of this. It's NON-NEGATIVE But it's also NON-POSITIVE These are an infinite set of positive integers, also known as the "counting numbers". Integer.

And all of these are... Whole Number

which is automatically an... So if you have a NATURAL number

like 42

It is always a..... These are also thought of simply as the Real numbers that aren't Rational. When written as a decimal Irrational numbers have

INFINITE and NON-REPEATING decimal expansions. 2.5482039547569302298182834950670121200091234857692154522908888234231234567809... Properties. History. Other Stuff.

Full transcript5

OR

2.97

OR

14567.3900723458679490202981834 Integers Rational Numbers 34343434343434 343434...FOREVER Whole Numbers Another way to say "fraction" is ratio. Hence the name RATIOnal numbers. Let's say this box represents all the real numbers. One special subset of rational numbers it the set of can never be the same as The Real numbers can be broken into two disjoint sets. Irrational Numbers do not include any 343434343434 Disjoint means non-overlapping. So something in one set cannot ever be part of the other. There is no such thing as a number that is BOTH rational and irrational, the sets are disjoint. which are not the same as All of the Real numbers that CANNOT be written as a fraction of two integers. A subset of the Integers is the set of Natural Numbers are disjoint from Take just the ZERO out of the Whole Numbers and you are left with the set of May not seem like much, but this turns out to be a really big deal. Real Numbers Rational Numbers Real numbers have two basic properties What are they? What is a FIELD???? Most cows are truly outstanding in their fields! Most cows are truly outstanding in their fields! Most cows are truly outstanding in their fields! They are an ORDERED field. They have a Least Upper Bound property. MEANS: Every number in the set can

be put in order from smaller to larger. Most cows are truly outstanding in their own field. No, not that kind of field.

Look it up! What is a mathematical

field? Report what you find. HUGE! Let’s say we have a set from A to B on the real number line. This set contains all numbers between A and B (doesn’t matter if A and B are in the set or not.) See all these points that are above B?

These are upper bounds for the set. This property just guarantees that there is one of these upper bounds somewhere that is smaller than all the rest… the Least of the Upper Bounds. Huh?!? (What kinds of Real Numbers are there?) Classification. Then what is out here? Any real number that can be written as a fraction of integers. Taken together, these two properties tell us that

the real numbers can be put into a single ordered line

and that the line has no holes in it anywhere. These

are critical features of the... 5.222222222222222222222.....

and

322.773216543654365436543..... OR decimal numbers that go on forever but at some point start repeating themselves. Like... In the box representing Rational Numbers there are some basic subsets. Integers are rational numbers with a denominator of 1

(so they can be written without using a fraction) -5 -123 -1 -34 -1000 0 1 2 3 4 5... 120 -8 zero 18 12,000,003 47 24 246 These can be described as the NON-NEGATIVE Integers. Why can't we make it simple and call them the POSITIVE Integers? Because of this. It's NON-NEGATIVE But it's also NON-POSITIVE These are an infinite set of positive integers, also known as the "counting numbers". Integer.

And all of these are... Whole Number

which is automatically an... So if you have a NATURAL number

like 42

It is always a..... These are also thought of simply as the Real numbers that aren't Rational. When written as a decimal Irrational numbers have

INFINITE and NON-REPEATING decimal expansions. 2.5482039547569302298182834950670121200091234857692154522908888234231234567809... Properties. History. Other Stuff.