Understanding what rational numbers and irrational numbers are

Rational numbers can be expressed as the ratio of two integers. Irrational numbers are the opposite—they cannot be expressed as the ratio of two integers. Rational numbers and irrational numbers are called the real numbers. The complex numbers are the real numbers plus the imaginary numbers (expressed in the form a+bi, where a and b are real numbers and i is an imaginary number). The imaginary number i is the square root of minus one. 

Rational numbers are either integers or fractions. They can be either positive or negative.  Examples are -1, 10, ¾, 7/8, … . Rational numbers that are fractions are either a culminating decimal or a repeating decimal. Any decimal representation that does not have a repeating pattern or terminate is an irrational number. The square root of a square is rational because it is an integer. Examples are (25)^1/2=5, (49)^1/2=7, (121)^1/2=11. However, the square root of any integer that is not a square is an irrational number. Examples of irrational square roots are 2^1/2, 3^1/2, 5^1/2, 6^1/2, 7^1/2, 8^1/2, 10^1/2, 11^1/2, 12^1/2, 29^1/2, 57^1/2, 133^1/2, etc. Examples of rational square roots are 4^1/2, 16^1/2, 25^1/2, 36^1/2, 100^1/2. These square roots are rational because they equal an integer after the square root is taken (2, 4, 5, 6, and 10 respectively). All square roots of integers that are not the square roots of a square integer have infinite non repeating decimals because they are irrational.  

An example of a repeating decimals is 1/7=0.142857142857…, the .142857 repeats forever. The fraction 1/5 is a terminating decimal because it is equal to 0.200000… .  The zeros never end (ad infinitum). The decimal representations of irrational numbers do not terminate or repeat. 

Two famous examples of irrational numbers are pi and e (Euler’s number). Pi is used to find the circumference and area of a circle. It is equal to 3.141592653589793… . The number e is the base of natural logarithms and is very useful in higher mathematics. It is equal to 2.71828182845904523536… . Whether or not e+pi is irrational is an unsolved problem of mathematics. The number e is 1 plus the reciprocals of the factorials of the nonzero positive integers. A factorial is all nonzero positive integers multiplied together up to the factorial integer. It can represented with an exclamation point, where n!=1x2x3x4x…xn. Examples are 3!=1x2x3=6 and 6!=1x2x3x4x5x6=720.  Pi can be represented using continued fractions, integrals, limits, products, series representations, etc. 

The irrational number the square root of 2 can be represented using a right triangle (triangle with one angle equal to 90 degrees). The side opposite the angle equal to 90 degrees is called the hypotenuse. The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides (c^2=a^2+b^2). If the two sides are equal to 1, then the hypotenuse is the square root of two because since 1^2+1^2=c^2=2, c=(2)^1/2. The square root of two is the length of this hypotenuse. This can be placed on the number line as follows: 

One unit is measured the same length as the two one unit sides of the right triangle just mentioned. The square root of two is the length of the hypotenuse of this triangle. Another example of an irrational hypotenuse is a right triangle with sides 1 and 2. The hypotenuse is 1^2+2^2=5=c^2, c=(5)^1/2.  Since the integer 5 is not a perfect square, it is an irrational number. 

The rational integer 3 is plus three units from zero on the number line. The rational number 1/3 is a little harder to locate. It is of course 1/3 of a unit from zero (to the right), but for an exact measurement more work is needed. Dividing 3 into 1 will get 0.333333… This means adding 0.3+0.03+0.003+0.0003+.00003+… will get any desired accuracy for the location of 1/3 on the number line. Evidently, irrational numbers can be located using a similar method. For example, the irrational number e is the sum of the reciprocals of the factorials. So, the representation in integers is 1/1+1/1+1/2+1/6+1/24+1/120+… This means the position of e can be found using the sum of these integers.

Rational numbers are easily expressed as the ratio of two integers. Irrational numbers are much harder. Important constants like pi and e have been proven irrational. Rational numbers can be located on the number line by measuring the distance from zero on the number line. A similar method can be used to measure the distance of known irrational numbers from zero on the number line.