Yo—Shmoop, represent.

You may not realize it, but you are already quite good at recognizing a variety of different symbols that all represent, or stand for, the same number or concept. For example, the item in parentheses (4) is not the number four. It is simply a symbol that represents the *concept* of the number four. To further illustrate this point, take a gander at the below:

4, 4, 4, 4, 4, 4, 4

Each of these symbols is different, albeit similar, but we can instantly identify each one of them as symbolic representations of the number four. Same deal with the Roman numeral "IV." So if a doctor ever tells you he's about to hook you up to an IV, you can always ask him if that's really necessary, or if he can instead just hook you up to a III.

Along the same lines, an amount can be expressed by either a fraction, decimal or percentage. For example, the underlying amount represented is the same whether we write or 75%, or draw this picture.

In each case, the symbol represents having three out of four equal parts, or 75% of the whole. If this were a football game, we'd be ready to head into the fourth quarter. If this were a dollar, we'd be 25 cents shy of being able to dry our laundry.

Rational numbers are often represented by fractions. Real numbers are usually represented as decimals. Percents are a particular way to represent fractions with a denominator of 100.

When given a particular problem or situation, choose the representation that best goes with the situation. When working with money, $0.75 usually makes more sense than "three-fourths of a dollar." When running a poll, "three-fourths of those polled" makes more sense than saying "0.75 of those polled." If you're saying that "three-fourths of those polled feel that $0.75 a week is enough to live on," then you could also say that there is a "100% chance that the polling methods are flawed."

Writing real numbers as decimals allows us to separate the real numbers into rational and irrational numbers. This is important, as they will often start to tussle when left alone together.

Rational numbers are decimals that end or repeat. Here are some examples of decimals that are rational numbers:

Irrational numbers are decimals that go on forever without repeating. That's pretty amazing when you think about it, considering that most of our politicians can't go more than about two minutes without repeating.

We can find many, many digits of irrational numbers, but we will never know them all:

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