If Jordan’s number is −32, and you have already figured out that −33 is too low and −31 is too high, then you know the answer is −32. You guess, and by going higher and lower you get closer and closer to the target. You decide to take the bait, and you quickly discover that this does not change the game much. Feeling pleased that you are getting closer, you ask, “How about 75?” “You got it!” Jordan replies, and you march triumphantly off to your first class of the day.īut after class, you again run into Jordan, who has apparently been thinking about ways to stump you: why stick to positive numbers? What if you also allow negative numbers? “Now I am thinking of a number between negative 100 and 100,” Jordan says gleefully. On Monday morning, your friend Jordan walks up to you and says, “I’m thinking of a number between 1 and 100.” Being a good sport, you play along and guess 43. What makes fractions so special? We explore how we can recognize the decimal representation of fractions and how fractions can be used to approximate any real number as closely as we wish. Centuries later, while we regularly use numbers that cannot be written as fractions, those numbers that can be written as fractions remain powerful tools. These number lines show that all integers are real numbers, but not all real numbers are integers.Legend has it that the first person in ancient Greece who discovered that there are numbers that cannot be written as fractions was thrown overboard from a ship. The blue line shown on top of the number line shows that all the values between the integers are included as well, not just their individual points. The number line below represents all real numbers. The number line below represents integers shown using red points to show that only whole values (not fractional or decimal values) are included in the set of integers. Real numbers and integers can be compared using number lines. Examples include π, Euler's number e, and the golden ratio. For example, ⅓ repeats indefinitely:Īn irrational number is made up of all the real numbers that are not rational numbers: non-terminating decimals that do not repeat. Decimals that repeat are indicated by writing a horizontal bar above the portion of the decimal that repeats. Rational vs irrational numbersĪ rational number is a number that can be expressed as a fraction where the numerator and denominator are integers, the ratio of which results in a terminating decimal, or a non-terminating decimal that repeats. The above is just a small sample of the various types of numbers that make up the real numbers.
Below are a few examples of real numbers.
The set of real numbers is indicated using this symbol: ℝ. Imaginary numbers are the result of trying to take the square root of a negative number. Real numbers were created to distinguish the set of real numbers from imaginary numbers. Real numbers are divided into rational numbers and irrational numbers, which include all positive and negative integers, 0, and all the fractional and decimal values in between ( fractions, decimals, transcendental numbers, etc.) Real numbers are one of the broadest categories of numbers.
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