Fractions

Fractions are usually thought of as ½, ¾, or maybe the decimal fraction 0.5. Perhaps some think of a fraction as a part of a circle or 5 parts out of 10 parts. The word "fraction" itself is related to a word meaning broken, as in "fracture."

Historically, fractions have been thought of and written in many different ways. Apparently the Babylonians, about 4,000 years ago, wrote fractions in a way similar to our decimal fractions, but instead of our base-10 , they used base-60 . They also used a space instead of a decimal point. This relatively simple system, however, does not reappear in common usage until about 3,600 years later in Europe.

Understanding Fractions

One mathematics dictionary defines a fraction as "a number less than 1." But this definition is too simple. The number is certainly not "less than one," but most people would still call it a fraction. Indeed, this same dictionary calls an important fraction or "a fraction whose numerator is larger than the denominator."

Another book defines a fraction as "a numeral representing some part of a whole." But what whole is representing part of, according to this definition? If a circle or a rectangle is divided into three equal parts, the denominator, or bottom, of the fraction is 3. But how many of those parts, or thirds, are represented by the fraction ? The numerator, or top, of the fraction tells you that represents five of the parts, each of which is one-third of a whole.

Another way to define a fraction is as a number that can be expressed in the form , where a and b are whole numbers and b is not equal to 0. But this definition, taken from another mathematics dictionary, has problems too. The whole numbers are 0, 1, 2, 3, 4, 5…. So does this definitionmean that -⅔ is not a fraction? Although -2 is not a whole number, clearly -⅔ is a fraction.

A better definition—and one that holds up under scrutiny—is that a fraction is a numeral written in the form where a can represent any number and b can represent any number except 0. By this definition, is a fraction, as are , , and as well as . This definition names a fraction according to the form. This means that is a fraction, and 50 is not a fraction. The values of the two numerals are the same, but the form of the numerals is not.

However, it is useful to be able to talk about the numbers that can be expressed as the ratio, or quotient, of two integers , such as ⅔, , and . These numbers, which can be negative, are called rational numbers . All rational numbers can be written in the form of fractions; however, not all fractions are rational numbers. For example, the fraction is not a rational number because π is not an integer.

see also Decimals; Form and Value; Fraction Operations; Numbers, Irrational; Numbers, Rational; Ratio, Rate, and Proportion.

Lucia McKay

Bibliography

Eves, Howard. An Introduction to the History of Mathematics. New York: Holt, Rinehart and Winston, 1964.

Hogben, Lancelot. Mathematics in the Making. London: Crescent Books, 1960.

———. The Wonderful World of Mathematics. New York: Garden City Books, 1955.

WHOLES AND PARTS

The Greek mathematicians (around 600 b.c.e. to 600 c.e.) had some problem working with fractions because the number 1 had a mystical significance as an indivisible unity. They did not want to break unity into parts, so they used ratios instead.