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Representing Subsets of Real Numbers Symbolically

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we will organize real numbers into subsets and show how this idea relates five types of numbers. Using various representations, we will learn how to conceptualize the subsets of real numbers.

Number Soup

Fred loves to visit his grandmother, Florene. She is not your typical grandmother, having retired from a very active technology career. This has impacted her domestic life. Instead of making alphabet soup, she cooks all sorts of number soups.

Using Florene's soup creations, we will explore ways to represent subsets of real numbers.

Sets of Numbers and Subsets

Florene starts her soup concoctions with naturally-shaped noodles. These natural numbers include 1, 2, 3, … and go on to infinity. They are the counting numbers. She calls this her ''Natural Set'' soup or just N for short. Fortunately, Florene has a very large kettle for this uncountable set of numbers.


Set of natural numbers, N
natural_number_set


By adding one ingredient, Florene creates a whole new soup called her ''Whole Set'' soup; abbreviated W. By simply adding 0 to the natural numbers we have the whole numbers: 0, 1, 2, 3, … We see N is a subset of W because N is contained W. This soup is quite wholesome.


Set of whole numbers, W
whole_number_set


Another box of special ingredients make this soup even more interesting. Florene adds negative natural numbers to the mix. Her soup now has integer numbers: {…, -3, -2, -1, 0, 1, 2, 3, …}. This is her ''Integer Set'' soup; also known as Z. Why Z and not I you ask? Well, I is reserved for the irrational set, which appears later in this cooking class. Do you see how Z contains the set of whole numbers? Right, W is a subset of Z. The compact way to write this uses the symbol ⊂ which means is a subset of. Thus, NWZ. Numbers enclosed by curly braces is another way to represent sets. Thus, {1, 2, 3, …} ⊂ {0, 1, 2, 3, …} ⊂ {…, -3, -2, -1, 0, 1, 2, 3, …}. That's right! A number like 2 is a natural number as well as being a whole number as well as being an integer. How about the number 0? Okay, 0 is a whole number and an integer but 0 is not a natural number.

Florene's ''Integer Set'' soup looks like:


Set of integer numbers, Z
integer_number_set


There is still a rational box of ingredients. Any number which can be written as a/b is in this box (a and b are integers). In this box are fractions (both proper and improper fractions), mixed numbers, and decimals. For example, 2/3, 3/2, 2.5 and -22 1/5. How about 1 and 0? Well, if a and b are the same, then a/b is 1. And 0 happens for any value of b (other than 0) provided a is 0. For example, 256/256 = 1 and 0/256 = 0. These ingredients are the rational numbers (a and b are in a ratio, which is a good way to remember ''rational''). And the letter Q is used for rational numbers. What! Why not R? Well, R is saved for the set of real numbers.

What about subsets?

NWZQ

Can all integers be called rational numbers? Ans.: Yes, we can write any integer as a/b.

Are there any rational numbers which are not integers? Ans: Yes. For example, 2/3 is a rational number but not an integer.

Florene's ''Rational Set'' soup:


Set of rational numbers, Q
rational_number_set


There is only one more ingredient missing to complete Grandma's ''Real Set'' soup. We need irrational numbers. Although, they look like decimals, these numbers can't be written as a ratio a/b. In fact, these decimals don't terminate, don't have a repeating pattern, and just go on to infinity. Want some examples? The number π is 3.1415926… Note, we approximate π with 22/7 but this is just an approximation. Other irrational numbers are e, the base of the natural logarithm. e = 2.718281… and √2 = 1.41421… By the way, for some reason, Fred really likes irrational numbers.

The set of real numbers, R, includes both the rational numbers and the irrational numbers. Both Q and I are subsets of R. But Q and I have no numbers in common.


Set of real numbers, R
real_number_set


Using a Tree Idea

Besides Florene's soups, another way to conceptualize number sets is with a tree structure. It's really an inverted tree with the root at the top. So, at the top all by itself is R, the real numbers. Everything below R is a real number.

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