What is Fraction? — Mathematics

Definition 24.1 Primary & Middle School Mathematics · 第 24 章 — First Fractions

Cut a whole into equal parts and take some — that is a fraction:

\frac{3}{4} \quad \begin{array}{l} 3 \text{: how many parts are taken (the \emph{numerator});}\\ 4 \text{: how many equal parts the whole is cut into (the \emph{denominator}).} \end{array}

$\frac12$ is a half, $\frac13$ a third, $\frac14$ a quarter, $\frac{1}{10}$ a tenth.

Reading fractions on pictures. The parts must be equal: three unequal pieces do not make thirds! — Reading fractions on pictures. The parts must be *equal*: three unequal pieces do not make thirds!

示例

Example 24.2 (Fractions of a collection)

$\frac13$ of $12$ marbles: share the $12$ marbles into $3$ equal groups of $4$ ; one group is $\frac13$ , so $\frac13$ of $12$ is $4$ — and $\frac23$ of $12$ is two groups, $8$ .

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Definition 39.1 Primary & Middle School Mathematics · 第 39 章 — Fractions: First Steps

Cut a unit into $b$ equal parts, and take $a$ of them: the quantity obtained is the fraction

\frac{a}{b} \qquad \begin{array}{l} a \text{: the \emph{numerator} --- how many parts we take;}\\ b \text{: the \emph{denominator} --- into how many parts the unit is cut.} \end{array}

Two fractions of the same bar. The denominator tells how fine the cutting is, the numerator how much we take.

示例

Example 39.2

A fraction can be bigger than $1$ : $\frac74$ means seven quarters — one whole unit (four quarters) and three quarters more:

\frac74 = 1 + \frac34 .

Example 39.5

Three pizzas for four people: each person gets $\frac34$ of a pizza. Check with the theorem: $4$ shares of $\frac34$ make $\frac34 \times 4 = 3$ pizzas. Some fractions are decimal numbers ( $\frac34 = 0.75$ ), others are not: $\frac13 = 0.333\dots$ never stops. The fraction is the exact value; $0.33$ is only an approximation.

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Definition 63.1 Primary & Middle School Mathematics · 第 63 章 — Fractions and Powers

A fraction $\dfrac ab$ (with $b \neq 0$ ) is the quotient of $a$ by $b$ . Two fractions are equal when one is obtained from the other by multiplying (or dividing) numerator and denominator by the same nonzero number:

\frac ab = \frac{a \times k}{b \times k} \qquad (k \neq 0).

Why a common denominator: cutting both wholes into 12 equal boxes makes the fractions comparable and addable — together, 10 + 9 = 19 twelfths. — Why a common denominator: cutting both wholes into $12$ equal boxes makes the fractions comparable and addable — together, $10 + 9 = 19$ twelfths.

示例

Example 63.2

Simplify $\dfrac{42}{56}$ step by step:

\frac{42}{56} = \frac{42 \div 2}{56 \div 2} = \frac{21}{28} = \frac{21 \div 7}{28 \div 7} = \frac34 .

(In Chapter 64 the greatest common divisor will do this in one step.)

Example 63.4

Compute $\dfrac56 + \dfrac34$ . A common denominator of $6$ and $4$ is $12$ :

\frac56 + \frac34 = \frac{5 \times 2}{6 \times 2} + \frac{3 \times 3}{4 \times 3} = \frac{10}{12} + \frac{9}{12} = \frac{19}{12}.

Compute $2 - \dfrac37$ . Write $2$ as a fraction over $7$ :

2 - \frac37 = \frac{14}{7} - \frac37 = \frac{11}{7}.

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