Comparing 2-digit numbers
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Most of us are familiar with the equal sign from our earliest days of arithmetic. You might see something like 1 plus 1 is equal to 2. Now, a lot of people might think when they see something like this that somehow equal means give me the answer. 1 plus 1 is the problem. Equal means give me the answer and 1 plus 1 is 2. That's not what equal actually means. Equal is actually just trying to compare two quantities. When I write 1 plus 1 equals 2, that literally means that what I have on the left hand side of the equal sign is the exact same quantity as what I have on the right hand side of the equal sign. I could have just as easily have written 2 is equal to 1 plus 1. These two things are equal. I could have written 2 is equal to 2. This is a completely true statement. These two things are equal. I could have written 1 plus 1 is equal to 1 plus 1. I could have written 1 plus 1 minus 1 is equal to 3 minus 2. These are both equal quantities. What I have here on the left hand side, this is 1 plus 1 minus 1 is 1 and this right over here is 1. These are both equal quantities. Now I will introduce you to other ways of comparing numbers. The equal sign is when I have the exact same quantity on both sides. Now we'll think about what we can do when we have different quantities on both sides. So let's say I have the number 3 and I have the number 1 and I want to compare them. So clearly 3 and 1 are not equal. In fact, I could make that statement with a not equal sign. So I could say 3 does not equal 1. But let's say I want to figure out which one is a larger and which one is smaller. So if I want to have some symbol where I can compare them, where I can tell, where I can state which of these is larger. And the symbol for doing that is the greater than symbol. This literally would be read as 3 is greater than 1. 3 is a larger quantity. And if you have trouble remembering what this means-- greater than-- the larger quantity is on the opening. I guess if you could view this as some type of an arrow, or some type of symbol, but this is the bigger side. Here, you have this little teeny, tiny point and here you have the big side, so the larger quantity is on the big side. This would literally be read as 3 is greater than-- so let me write that down-- greater than, 3 is greater than 1. And once again, it just doesn't have to be numbers like this. I could write an expression. I could write 1 plus 1 plus 1 is greater than, let's say, well, just one 1 right over there. This is making a comparison. But what if we had things the other way around. What if I wanted to make a comparison between 5 and, let's say, 19. So now the greater than symbol wouldn't apply. It's not true that 5 is greater than 19. I could say that 5 is not equal to 19. So I could still make this statement. But what if I wanted to make a statement about which one is larger and which one is smaller? Well, as in plain English, I would want to say 5 is less than 19. So I would want to say-- let me write that down-- I want to write 5 is less than 19. That's what I want to say. And so we just have to think of a mathematical notation for writing "is less than." Well, if this is greater than, it makes complete sense that let's just swap it around. Let's make, once again, the point point towards the smaller quantity and the big side of the symbol point to the larger quantity. So here 5 is a smaller quantity so I'll make the point point there. And 19 is a larger quantity, so I'll make it open like this. And so this would be read as 5 is less than 19. 5 is a smaller quantity than 19. I could also write this as 1 plus 1 is less than 1 plus 1 plus 1. It's just saying that this statement, this quantity, 1 plus 1 is less than 1 plus 1 plus 1.