# Inequalities and Equivalence: Less-than and Greater-than

Topics: Science

## Less-than and Greater-than

We're going to start with inequalities, beginning with a short introduction. This lecture
will give you the basic idea behind symbols like < b, x > y and c ≤ d. The first four of
these are really well-defined mathematical concepts.
Inequalities, Basic Ideas
a<b
c<=d
x<y
z=>w
e<<f
Let's start from the top and draw a real number line. And suppose here is 0, here is
the number 2. here is the number 3.1, and let us consider this statement: 2 < 3.1.
This funny little pacman symbol here pointing that way means "is less than." So here
we have a statement; let us think about what it means and see why it is true. This
statement: 2 < 3.1 means "2 is to the left of 3.

1 on the real number line." Let's say
11.78 < 3.1; say it's about there. Fact: 11.78 is also to the left of 3.1 on the number
line (on a real number line). In general when we write A less than B, we mean A
wherever it is on the real number line (on a real number line).
The symbol < denotes less than. There is no universally accepted symbol for greater
than. It can be written as >, or 2, or sometimes it is not written at all. When two
numbers are compared using S, it means that the first number (on the left) is less
than the second number (on the right). For example, 3.

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1 <2 means that 3.1 is less
than 2. The other comparison operators are: >, >=, and ≤ .
[Graph]
Here's this funny symbol, x << y. And you actually never see this in a proper math
textbook, because it's not really a proper mathematical concept. But you'll see this all
the time in data science. What this really means x is much, much less than y. So for
instance, the statement 1 < 1,000,000 might be a reasonable thing. I would argue
that it is not true. It is not possible for a judge to determine whether this statement is
true or false. Whereas a statement like 2 < 3.1 can be determined to be either true or
false. It is much less than. It's in the eye of the beholder but we tend to agree on
what it means. Thus far we have considered statements with greater than and less
than statements: now let's think about equal to statements.
[Graph]
To write the mathematical symbol "<=" as "less than or equal to," one should first
write the less-than sign and then place a horizontal line below it. One also
sometimes sees this symbol written as "a < b". This means that a <b, or a = b. It's
just shorthand way of saying that either a is less than b. or a is equal to b. If one
wants to write the sentence "a is less than or equal to b", it can be written two ways:
"a <b" or "a ≤b".
When a claim is made, you can test it by trying to disprove it by showing that one of
its statements is false, while the other is true. For example, suppose someone claims
that 2 < 3.1. This means that either 2 < 3.1, or 2 = 3.1. The former statement is false,
since 2 > 3 Therefore, the person's claim 1S true only if he meant the latter
statement and said that 2 = 3.
[Graph]
Suppose someone claims that 2 is less than or equal to 0.8. This means either 2 <
0.8 or 2 = 0.8. However, both statements are false and so the claim is itself false
overall.
This concludes the lecture on basic inequalities. We've learned what less than
means, greater than, less than or equal to, and greater than or equal to, all of which
are standard mathematical concepts. We've also learned what it means to say that
A <B, which is not a standard mathematical concept but one that people use
informally.