Section 24.4 Closing Ideas
The ability to see the same idea from multiple perspectives creates the opportunity to apply different approaches to solving problems. In this section, we saw two different ways to represent numbers.
The number line is a purely geometric framework. It turns out that the Greek mathematicians thought of numbers this way, but in an even more strict sense. If they wanted to compare the numbers and then they would (essentially) say that is longer than But how would this work with negative numbers? As it turns out, the Greeks never really bothered with negative numbers. To them, they didn't exist because they only worked with counting numbers and lengths. Since the number line contains negative numbers, we will have an additional set of tools that the Greeks did not have when it comes to thinking about numbers and mathematical ideas.
The base-10 blocks gives us another geometric framework, but it ties more closely with our sense of how we represent numbers instead of giving us insights into the numbers themselves. It may seem obvious to us because we've worked with numbers this way our whole lives. But not every system of writing numbers uses a place value system. For example, Roman numerals are notoriously difficult for students to learn because it's built around rules that are not always intuitive and sometimes feel arbitrary.
As you come to understand more mathematics, you can start to develop a more flexible mindset for looking at ideas. This can often lead to new insights and more interesting questions.