Understanding the Mathematical Subjects Decimals and Fractions

A decimal is equivalent to a fraction when the decimal is equal to the numerator of the fraction divided by the denominator. Examples are 0.75=3/4 because four divided by three equals seventy-five hundredths, 0.3=3/10 because ten divided by three equals three tenths, 0.5=1/2 because one divided by two equals five-tenths, and 0.25=1/4 because one divided by four equals twenty-five hundredths. 

Mixed numbers work the same way.  Examples are 2.33=2 33/100 because thirty-three divided by one hundred equals 0.33 and adding two makes it 2.33, 3.8=3 4/5 because four divided by five equals 0.8 and adding three makes it 3.8, and 100.91=100 91/100 because ninety-one divided by one hundred equals 0.91 and adding one hundred makes it 100.91.

It can get more complicated. There is a good way to determine if a fraction is rational or not. A rational number is any number that can be expressed by a ratio of two numbers called a fraction. A number that is not rational is called irrational. A repeating decimal is a fraction if it terminates or has a repeating pattern.  Examples of terminating decimals are 0.333, 0.78490, 0.5457, 0.5, 0.28, etc. They are easily proven rational by using the numbers in the decimal as the numerator and the place of the decimal for the denominator. This makes 0.333=333/1000, 0.78490=78490/100000, 0.5457=5457/10000, 0.5=5/10, and 0.28=28/100.

The decimals with repeating patterns are harder to prove rational.  Common fractions like 1/3, 1/7, and 1/9 have repeating decimals. The fraction 1/3 is a repeating decimal because 1/3=0.3333… . The 3 three repeats infinitely. And the fraction 1/7 is a repeating decimal because 1/7=0.142857142857142857142857… . The 142857 repeats infinitely. Also the fraction 1/9=0.1111111… . The one repeats infinitely.  

Converting a decimal to a fraction is harder than converting a fraction to a decimal. It is done by dividing the integers of the decimal by the corresponding places of the decimal. One example is 0.378=3/10+7/100+8/1000=300/1000+70/1000+8/1000=378/1000=0.378.

One-thousandth is the common multiple of tenths, hundredths, and thousandths, so tenths are multiplied by one-hundredth, hundredths by one-tenth, and thousandths are multiplied by one. Another example is 0.7893=7/10+8/100+9/1000+3/10000=7000/10000+800/10000+90/10000+3/10000=7893/10000=0.7893.

Irrational numbers are numbers that are not rational, that is, they cannot be expressed as the ratio of two integers. The square root of an integer is rational if it is a perfect square, but is irrational otherwise. It can be easily seen that the square root of a perfect square is an integer and therefore rational (integers are rational because they can be expressed as the ratio of the integer divided by one). Examples are (9)^1/2=3 because 3×3=9, (16)^1/2=4 because 4×4=16, and (144)^1/2=12 because 12×12=144. Interesting irrational numbers are pi and e.

Pi was first used to find the circumference and area of a circle. The formulas are the circumference equals pi times the diameter and the area equals pi times the radius squared. Pi is equal to 3.141592653589793… to the first fifteen decimal places.

The number e is more complicated, but one of the most important numbers in mathematics. It was named after Euler. It can be found many ways, one of the most famous is e is equal to the sum of the reciprocals of the factorials starting with zero. A factorial is all the integers up to and including the integer of the factorial multiplied together. Examples are three factorial equals 1x2x3=6 and 5 factorial equals 1x2x3x4x5=120. The factorial of zero is defined to be one. So, e is 1/1+1/1+1/2+1/6+1/24+1/120+… It equals 2.71828182845904523536… to the first twenty decimal places.