What is the meaning behind the common fraction notation $\frac{a}{b}$?

Question

Why is the dash sign "-", which indicates division, used in common fractions? Is it because $a$ is divided by $b$? If we understand division in a literal sense, for example, $1$ cake must be divided into $4$ parts, it is quite clear that the fraction $\frac{1}{4}$ means a “quarter” of the cake. Then what does $\frac{3}{4}$ of the cake mean? Or is it just a shorthand way of writing (where the denominator is the number of equal parts into which something is divided and the numerator is the number of taken parts), for example, “two thirds of a rope” is $\frac{2}{3}$? Can someone explain the meaning behind this notation with examples?

Answer 1

The little dash is called a vinculum. I am going to refer to the dash by calling it "the vinculum"

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Basic Definition

Keep in mind that the basic definition of a fraction $\dfrac{a}{b}$ is $a\div b$.

$\dfrac{1}{4}$ of a cake means $1$ cake divided by $4$. $\dfrac{26}{7}$kg means $26$kg divided by $7$.

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Historic Origin

From Earliest Uses of Symbols for Fractions:

The horizontal fraction bar was introduced by the Arabs. "The Arabs at first copied the Hindu notation, but later improved on it by inserting a horizontal bar between the two numbers" (Burton).

Several sources attribute the horizontal fraction bar to al-Hassar around 1200.

When Rabbi ben Ezra (c. 1140) adopted the Moorish forms he generally omitted the bar.

Fibonacci (c.1175-1250) was the first European mathematician to use the fraction bar as it is used today. He followed the Arab practice of placing the fraction to the left of the integer (Cajori vol. 1, page 311).

According to the DSB, Abu Abdallah Yaish ibn Ibrahim ibn Yusuf ibn Simak al-Umawi (14th or 15th century) insisted that the horizontal fraction bar be used, whereas easterners continued to write it without the bar.

Why Vinculum

Using the vinculum brings many benefits as compared to using a slash or divide sign.

1. Reduces Ambiguity.

The vinculum, in some way, "groups" the numerator and denominator together, which makes the fraction clearer. For example, $\ln\frac{a}{b}$ is indisputably clear. You are taking the $\ln$ of $\frac{a}{b}$. However, if you have $\ln a\div b$, it is not clear! Do you mean $\ln (a\div b)$ or $(\ln a)\div b$? The same is true for the slash sign. Does $\ln a/b$ mean $\ln (a/b)$ or $(\ln a)/b$?

To conclude, the first benefit of using a vinculum is that it "groups" the numerator and the denominator together, which improves clarity and reduces ambiguity.

2. Visual Clarity.

What the vinculum also does, is that it makes the equation much more aesthetically pleasing and much easier to understand. Suppose you have

$$\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\dots+\frac{1}{a_n}}\leq\frac{a_1+a_2+a_3+\dots+a_n}{n}$$

and

$$n/(1/a_1+1/a_2+1/a_3+\dots+1/a_n)\leq(a_1+a_2+a_3+\dots+a_n)/n$$

You tell me, which one is easier to understand?