Understanding Inequalities: Less-than and Greater-than Concepts

Less-than and Greater-than We're going to start with inequalities, beginning with a short introduction. This lecturewill give you the basic idea behind symbols like < b, x > y and c ≤ d. The first four ofthese are really well-defined mathematical concepts. Let's start from the top and draw a real number line. And suppose here is 0, here isthe 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 herewe have a statement; let us think about what it means and see why it is true. Thisstatement: 2 < 3.1 means "2 is to the left of 3.1 on the real number line." Let's say11.78 < 3.1; say it's about there. Fact: 11.78 is also to the left of 3.1 on the numberline (on a real number line). In general when we write A less than B, we mean Awherever it is on the real number line (on a real number line).The symbol < denotes less than. There is no universally accepted symbol for greaterthan. It can be written as >, or ≥, or sometimes it is not written at all. When twonumbers are compared using <, it means that the first number (on the left) is lessthan the second number (on the right). For example, 3.1 < 2 means that 3.1 is lessthan 2. The other comparison operators are: >, >=, and ≤ . Here's this funny symbol, x << y. And you actually never see this in a proper mathtextbook, because it's not really a proper mathematical concept. But you'll see this allthe time in data science. What this really means x is much, much less than y. So forinstance, 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 istrue or false. Whereas a statement like 2 < 3.1 can be determined to be either true orfalse. It is much less than. It's in the eye of the beholder but we tend to agree onwhat it means. Thus far we have considered statements with greater than and lessthan statements; now let's think about equal to statements. To write the mathematical symbol "<=" as "less than or equal to," one should firstwrite the less-than sign and then place a horizontal line below it. One alsosometimes sees this symbol written as "a < b". This means that a < b, or a = b. It'sjust a shorthand way of saying that either a is less than b, or a is equal to b. If onewants 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 ofits statements is false, while the other is true. For example, suppose someone claimsthat 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 is true only if he meant the latterstatement and said that 2 = 3. 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 falseoverall. This concludes the lecture on basic inequalities. We've learned what less thanmeans, greater than, less than or equal to, and greater than or equal to, all of whichare standard mathematical concepts. We've also learned what it means to say thatA << B, which is not a standard mathematical concept but one that people useinformally.