Irrational Numbers
Peter Oye Sagay
Irrational numbers are numbers that can not be expressed as ratios of integers. In other words, the decimal part of irrational numbers repeat without end and without a repetitive pattern. Irrational numbers together with rational numbers (numbers that can be expressed as ratios of integers) form the real number line.
Ancient Babylonians and Egyptians used irrational numbers in problem solving. However, they used the numbers as approximations
rather than as a distinct group of numbers. The discoverer of the distinctiveness of irrational numbers was a mathematician who was a member of the great group of
mathematicians called the Pythagoreans. This group was founded by Pythagoras, the great mathematician who lived about 582BC - 500BC. The Pythagoreans made several
great discoveries in music, astronomy, geometry and number theory. Numbers were a central part of their philosophy. So much so that they believed that rational
numbers were at the heart of all phenonmena. The Pythagoreans were particularly interested in square numbers. A square number is a number that
is equal to a number raised to a power of 2. For example, 4, 9, 16 and 25 are square numbers
because 4 = 2 2 = 2 x 2; 9 = 32 = 3 x 3; 6 = 42 = 4 x 4 and 25 = 52 = 5 x 5.
As a result of their research, the Pythagoreans discovered that the sum of
certain square numbers are also square numbers. For example, the sum of the square numbers 9 and 16 is the square number 25.
In mathematical terms, 32 + 42 = 52. The numbers 3, 4 and 5 in this equation are called Pythagorean Triples.
In general, a, b and c are Pythagorean Triples if a2 + b2 = c2. The Pythagoreans extended their study of
square numbers into geometry. This endeavor resulted in the geometric theorem now known as the Pythagorean Theorem: the sum of the squares of
the sides of a right angled triangle is equal to the square of the hypotenuse. The discovery of the Pythagorean Theorem, provided the platform
for the discovery of irrational numbers as a distinct group of numbers.
One day, a Pythagorean decided to calculate the length of the hypotenuse of a right angled triangle when the other two sides are equal to one. Using the Pythagorean theorem, he found that the square of the length of the hypotenuse is 2, that is, 12 + 12 = 2. Since 2 is not a square number, the length of the hypotenuse cannot be a whole number. Therefore, this Pythagorean surmised that there had to be a rational number whose square is equal to 2 since the Pythagoreans believed that all numbers can be reduced to rational numbers. He decided to investigate and he found that √2 was not a rational number. This discovery was not great news for the Pythagoreans who had built a system of philosophy based on rational numbers. So they decided to keep the discovery secret.
The problem with secrets involving many people is that sooner or later they are revealed. Greek mathematicians of the period, soon got to know that √2 is not a rational number and they later discovered that there are many other numbers that are not rational numbers: the square root of numbers that are not perfect squares; the cube root of numbers that are not perfect cubes, the constants pi (π), e and many other numbers. These numbers were called irrational numbers.
It took many centuries before irrational numbers were firmly embraced in the mathematics community. Mathematicians acknowledged their relevance as measures of geometric lengths and areas, but beyond this, there was little or no interest. The scientific progress and mathematical rigor of the 17th century brought some irrational numbers into the limelight, most notably, pi and e. Consequently, there was renewed interest in the characteristics of irrational numbers and their arithmetic.
The Arithmetic of Irrational Numbers
In general, the following are true given the irrational numbers √a and √b, where a and b are not perfect squares:
√a + √b = √a + √b
√a - √b = √a - √b
√a x √b = √ab
√a/ √b= √(a/b)
The square root symbol, √, is called the radical and the numbers
a and b the radicand.
The Usefulness Of Irrational Numbers
Two of the most important mathematical constants, pi (π) and e, are irrational numbers.
These mathematical constants are used in many branches of mathematics: geometry, growth equations,
calculus, current theory etc. For example, pi = the ratio of the circumference of a circle to its diameter. pi is also
used to calculate the surface areas and volumes of many geometric forms. The constant e is used as the limit of
certain infinite series; as the base for the naperian or natural logarithm; as a probability distribution; etc.
In trigonometry, the cosine and sine of 45o are respectively the reciprocal of the square root of 2, an irrational number.
The cosine of 30o and the sine of 60o are derived from ratios having the square root of 3, an irrational number, as numerator.
These are very important uses of irrational numbers.