Rational Numbers

The idea of rational numbers is a major step in abstraction in thinking about numbers. It moves from thinking about numbers as representing concrete physical situations to numbers as entities which can be operated on without any reference necessarily to physical concrete situations.

Greeks & Rational Numbers

The idea of rational numbers is a legacy of the Greeks. Greeks did not use numbers to directly measure lengths of line segments or angles. They referred to lengths as linear magnitudes.

Music is made of sounds & beats. The musicality of sounds (called notes) depends on the ratio of their frequencies of vibration! Pythagoras is supposed to have noticed this when listening to the sounds of iron pieces being hammered. He is also supposed to have noted that the notes whose frequencies were simple fractions were most melodious when sounded together. So he grandly declared that the entire universe is governed by numbers!

Greeks compared line segments in two ways - additively & multiplicatively. Additively they compared two line segments as more/ less/ equal to each other. Multiplicatively they compared them by taking their ratios.

They also developed this notion that the ratio of any two magnitudes can be represented as a ratio which can be expressed in terms of integers. Their notion was challenged by the discovery of irrational numbers!

From the idea of ratios, Greeks developed the idea of rational (derived from ratio) numbers which are expressible as a ratio of integers.

But it is modern mathematics which used this as a powerful idea to represent numbers.

Real Numbers are those which can represent quantities. They can be plotted on a number line. Real numbers can be divided into rational & irrational numbers.

Rational number is any number which can be written in the form a/b, where a & b are integers (numbers like -2) and b is not equal to 0. It is better if a & b are relatively prime.

An irrational number is one which cannot be represented in the form a/b, where a & b are integers.

When a rational number is converted to decimal notation the digits in the decimal portion start repeating after a certain number of digits. The digits which repeat are called "recurring" digits. There is an interesting reason for this.

The conversion to a decimal notation is done by a process of division. While dividing, the remainder at any stage is always less than the divisor. It can be any number from 0 to 9. If the division continues, the remainder number has to repeat at some point as it has only ten ( 0 to 9) possibilities. Whenever the remainder occurs again, the entire process of division repeats itself and the digits in the quotient start repeating.

In an irrational number, the digits in the decimal part never repeat. This is the reason they cannot be represented like rational numbers. The recurrence of digits is the major difference between a rational number and an irrational number.

Some life situations could be related to the idea of rational numbers, like comparing parts of a whole to the whole. But essentially, they are thought of as abstract entities which can be operated on as per certain rules accepted by all mathematicians.

The name ‘rational’ here indicates that the idea emerged from ratios which are numbers of the type a/b, where a & b are positive numbers. In a rational numbers they can be integers.

The rules of operating on rational numbers are worked out such that they follow the laws of Arithmetic. Any entity which obeys these laws is considered as a number.

Rational Numbers imply just the form in which the number is written. They lend themselves to multiple interpretations, of which idea fractions is just one. They are similar to the rules for operating on fractions.

Fraction- A fraction can be thought of a whole being divided into 8 equal parts out of which 5 are taken. This can be called the Part-Whole relation. A fraction cannot have a negative value.

Ratio- Another is the Ratio idea – as can be thought of as a ratio of a & b. The ratio idea is not a number in the sense we have studied in primary classes. It is more like a relation between a & b. This relation, as we shall see in Section 18 on ratios, is qualitatively different from the relation between a & b in a fraction of the same form. Ratios also cannot have a negative value.

Division fact-Another is the division idea where a/b represents the division of a by b. Numerically this leads to the same result. Dividing 1 pizza into 4 equal parts and taking 3 out of them leads to the same quantity as taking 3 pizzas and dividing them into 4 equal parts.

Measure- Another idea is that a rational number can be marked on a number line much like whole numbers. In that way it gives a measure of that number. A number like 3/5 will be located between 0 and 1. A number like 7/5 will be located between 1 & 2. A number like -3/5 will be located between 0 and -1.

Decimal- As an extension of the Measure idea, rational numbers can also be thought of as decimals. The decimal representation of rational & other numbers, brought out their differences very clearly and led to the idea of the set of real numbers.

The idea of rational numbers covers all the numbers we have studied until now (whole numbers & fractions) as well as numbers we would shortly be studying (decimal fractions & integers). It also sets the stage for learning about irrational numbers, which we would study in chapter 17.10.

In one sense, we can say that up to rational numbers, the idea of numbers was based on the idea of counting. From irrational numbers onwards the idea of numbers is based on geometry.