第1章基礎

Fractions are a way to represent parts of a whole. The fraction $\frac{1}{3}$ means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See Figure 1.11. The fraction $\frac{2}{3}$ represents two of three equal parts. In the fraction $\frac{2}{3},$ the 2 is called the numerator and the 3 is called the denominator.

If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate $\frac{6}{6}$ pieces, or, in other words, one whole pie.

So $\frac{6}{6}=1.$ This leads us to the property of one that tells us that any number, except zero, divided by itself is 1.

If a pie was cut in $6$ pieces and we ate all 6, we ate $\frac{6}{6}$ pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate $\frac{8}{8}$ pieces, or one whole pie. We ate the same amount—one whole pie.

The fractions $\frac{6}{6}$ and $\frac{8}{8}$ have the same value, 1, and so they are called equivalent fractions. Equivalent fractions are fractions that have the same value.

Let’s think of pizzas this time. Figure 1.12 shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that $\frac{1}{2}$ is equivalent to $\frac{4}{8}.$ In other words, they are equivalent fractions.

How can we use mathematics to change $\frac{1}{2}$ into $\frac{4}{8}?$ How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into $8$ pieces instead of just 2. Mathematically, what we’ve described could be written like this as $\frac{1·4}{2·4}=\frac{4}{8}.$ See Figure 1.13.

This model leads to the following property:

If we had cut the pizza differently, we could get

So, we say $\frac{1}{2},\frac{2}{4},\frac{3}{6},\text{and}\frac{10}{20}$ are equivalent fractions.