Rational numbers and fractions

A coin with 2 1/2 G as description. Dutch two and a half guilder coin form 1960.

Figure 1. Dutch two and a half (2½) guilder coin (from before the Euro era).

Fractional numbers

Earlier we left the remainder of a division undivided because a remainder is too small to divide it without cutting it in parts. Now let's see what happens if we allow cutting and cut the remainder in equal parts and evenly divide the parts:

14 ÷ 3 = 4 r 2 14 ÷ 3 = 4 + (2÷3) 14 ÷ 3 = 4+23

Instead of each of the 3 getting 4 and leaving 2 undivided, each now gets 4 plus a "part of one". This "part of one" or "fraction of one" cannot be expressed as an integer, it is a fractional or broken number. We represent this fractional number by simply maintaining the same notation as for the division, generally by using the slash or the horizontal fraction bar. So, in the example above 2/3 represents both an operation (2 ÷ 3) and a fractional number. Figure 2 shows how (a remainder of) 2 pizzas are divided over 3 persons.

Dividing a remainder 1 2 3 ÷ = 1/3 1/3

Figure 2. Two pizzas, evenly divided over three persons. Each person gets two slices of 1/3 of a pizza.

If 2 (remaining) pizzas are evenly divided over 3 persons, the 2 pizzas each need to be cut in 3 equal slices, and each of 3 persons will get one slice (= 1/3 pizza) of each of the two pizzas, which equals 2/3 pizza per person. The numerator 2 represents the number of equal slices, and the denominator 3 indicates how many of these slices make up a whole pizza and therefore the size of a slice.

Rational numbers

Fractional numbers are numbers "between" the integers, representing a part of a whole. The set of integers (positive, negative and zero) and the set of fractional numbers (positive and negative) together form the set of rational numbers (a rational number is a "ratio" of integers).

A rational number is a number that can be expressed as a quotient or fraction of two integers. In other words: a rational number is a number that can be expressed as the common fraction:

ab = a/b

in which a is the integer numerator and b is the non-zero integer denominator.

So again, the set of rational numbers include the fractional numbers (e.g. 2/3) and the integers (e.g. 3/1 = 3). The set of rational numbers is denoted by a capital blackboard bold ℚ, which stands for quotient (see fig.3).

Figure 3. The set of complex numbers ℂ includes the set of real numbers ℝ, which includes the set of rational numbers ℚ, which includes the set of integers ℤ, which includes the set of natural numbers ℕ.

Apart from (common) fractions, rational numbers can also be expressed as percentages or decimal fractions (both of which will be discussed in more detail later).

Rational numbers can be placed on a number line. The same way we earlier divided one pizza, we divide "one" (the line segment between two successive integers) on the number line in 3 equal parts. In the next picture the rational numbers 1/3, 2/3 and 4 + 2/3 are shown on a number line.

1 2 3 4 5 0 -1 1 3 2 3 4+ 2 3

Fractions

A fraction is an expression with a numerator and a denominator as presented above. The numerator and denominator do not have to be integers; they can also be fractions again or irrational numbers (e.g. π/4) or algebraic expressions. A fraction does not have to represent a rational number. A fraction may represent a ratio of rate (later more about this), a division or quotient or an algebraic fraction (quotient of algebraic expressions):

3 x 2 x2 + x

4 5 2 11 2 3 + 1 7 = 34 55 17 21 = 34 55 × 21 17 = 42 55

Common fractions

A common fraction (or simple fraction or vulgar fraction) is a fraction in which numerator and denominator are both integers. It can be used to write a rational number. A common fraction represents a number of equal parts per other number of equal parts. This can be a ratio or rate (later more about this), a division or quotient or a number of equal parts per number of these parts that make up a whole, as in the pizza example above.

For example, 42 55 is a common fraction, while 4 5 2 11 2 3 + 1 7 is not.

Proper fractions and mixed numbers

A common fraction between zero and 1 is called a proper fraction. In a proper fraction the numerator is always less than the denominator. A proper fraction represents a part of one (unit).

12 , 23 , 13 , 34 , 25 , 78 , 83100

Verbally expressed as: "a half", "two thirds", "one third", "three quarters" (or "three fourths"), "two fifths", "seven eighths", "eighty-three one-hundredths".

In the chapter about division we applied long division in a a few examples to find the quotient and remainder. When we also divide the remainder:

153 ÷ 4 = 38 r 1 153 ÷ 4 = 38+14

812 ÷ 5 = 162 r 2 812 ÷ 5 = 162+25

1707 ÷ 36 = 47 r 15 1707 ÷ 36 = 47+1536

13610 ÷ 15 = 907 r 5 13610 ÷ 15 = 907+515

162 163 162+ 2 5

Quotients in which the remainder is presented as a proper fraction are called mixed numbers. A mixed number is a representation of a rational number by an integer part together with a fractional part. The fractional part is a proper fraction, a part of one unit that each gets after dividing the remainder. The integer part is the integer closest to zero:

4+ 2 3

And not:

5 1 3

Mixed numbers are usually denoted without the plus or minus sign. This should not be confused with implied multiplication.

4+ 2 3 = 4 2 3

162+ 2 5 = 162 2 5

2 1 4 = 2 + 1 4 = 2 1 4

Proper fractions divide the unit between 0 and 1, but what happens if we divide the unit between any other two successive integers? Let's continue the graduation by dividing all the units on a number line in 3 parts (2 marks between any two successive integers):

1 2 3 0 -1 3 3 2 3 1 3 0 3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 10 3

Now improper or top-heavy fractions appear, where the numerator is equal to or greater than the denominator. We see that:

3 3 = 1 6 3 = 2 8 3 = 2 2 3

Improper fractions are easier to work with than with mixed numbers, but in the final result they will usually be converted to a mixed number (or to a decimal fraction, but this will be covered in a later chapter). This is often referred to as "reducing" or "simplifying" the fraction. Any improper fraction can be reduced to either an integer or a mixed number.

In fact, reducing a fraction is the same as dividing the numerator by the denominator, as we already know:

14÷3 = 14 3 14÷3 = 4 2 3