What is an Infinite Number?

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Mathematics can help illustrate the mind-bending concept of infinity, around since ancient times. Common numbers used in basic mathematics stretch on infinitely but fit neatly into simple equations. Using these numbers is an essential part of learning fractions in arithmetic, measuring circles in geometry and delving into advanced algebraic concepts such as square roots.

Identification

  • An infinite number, more accurately called an infinite decimal, indicates any number that contains an endless line of digits after a decimal point. For example, people often use the decimal notation 0.333 to indicate the fraction 1/3. Dividing one by three, however, actually produces an endless amount of 3s following the decimal. Similarly, the constant pi--the ratio of any circle's diameter to its circumference--carries an infinite number of digits following the usual estimation of 3.14.

Types

  • Two general types of numbers repeat infinitely. Numbers that repeat in a pattern--0.333..., 0.3888... or 7.185185185...--are rational numbers. You can write all of these numbers as fractions: 1/3, 7/18 and 7 5/27, respectively. Irrational numbers, such as the square roots of 2 and 3, continue on infinitely without ever falling into a repeating pattern. Mathematicians have mapped out pi to billions of digits without a pattern emerging. Note also that some infinite decimals can seem to follow a logical pattern--0.1010010001... for example--but these also are irrational numbers, because the digits themselves never repeat and you cannot write them as fractions.

Theories

  • Numbers that continue infinitely have properties that can seem contrary to their appearance. In a popular example, you can use mathematics theory to prove 0.999.... has the same value as 1. For instance, 1/3 equals 0.333..., and 2/3 equals 0.666... Adding those together as fractions equals 3/3, or 1. Adding the decimals together, however, equals 0.999... Similarly, the equation 1 - 0.999... gives the solution 0.000..., with an infinite number of zeroes that never reaches a 1, indicating they are of equal value.

Misconceptions

  • Infinity itself, symbolized by a figure that resembles a sideways 8, is not a number. You could write it in the format of an infinite number, such as a 1 followed by an infinite number of zeroes. This, however, is a concept, not a number. By definition, you cannot quantify it. Despite the popular one-upsmanship phrase "infinity plus one," you cannot add, subtract, multiply or divide infinity and get anything besides infinity.

Considerations

  • Although infinity itself is not a quantifiable number, there are both countable and uncountable infinities. For example, take two series of numbers: 1, 2, 3, 4.... and 1, 1.5, 2, 2.5, 3, 3.5, 4... While both series continue infinitely, the second series potentially contains twice as many numbers as the first series. You cannot quantify some broader sets, however, such as the amount of numbers between 1 and 2. This set would include 1.1, 1.11, 1.111 and infinite other number combinations.

References

  • Photo Credit pi image by Ewe Degiampietro from Fotolia.com
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