Diagram for Math NumbersDate: 10/05/97 at 22:44:33 From: Rina Hallock Subject: Help with Tree Diagram of math terms.... My daughter is doing a tree diagram using terms related to math "numbers" (for a calculus assignment). Could you please explain in lay terms what surds are? I don't remember ever having that term, and she can't find it in her resources. She has the branch with irrational numbers completed; we're working through the rational numbers. She has one branch off rational numbers started - whole numbers. Whole nubmers is branched to natural numbers and integers. I'm not sure that she has that right. The terms with which she is working include Real numbers, Surds, Opposites of natural numbers, Even numbers, Repeating decimals, Non integers, Natural numbers, Irrational numbers, Transcendental numbers, Integers, Zero, Rational numers, Whole numbers, Odd numbers, Terminating decimals. Any help you can give would be greatly appreciated. Thanks, Rina Hallock Date: 10/10/97 at 16:22:03 From: Doctor Chita Subject: Re: Help with Tree Diagram of math terms.... Dear Rina: Making a tree diagram of the various number systems is a nice way to show their relations - in particular, noting which sets of numbers are subsets of other numbers. For example, the natural numbers (also called "counting numbers) are a subset of the whole numbers. The natural numbers consist of the numbers {1, 2, 3, ... }, and the whole numbers consist of the numbers {0, 1, 2, 3, ... } The ellipsis (...) in each case indicates that the numbers go on forever, without bound. There is no subset of largest numbers. The list that your daughter's teacher gave her includes several redundancies. For example, you asked about "surds". A surd is another name for an irrational number. Thus, sqrt(2) is a surd. It is also an irrational number. This means that when you try to evaluate it as a decimal, it neither repeats nor terminates. A decimal number that repeats is 0.3333... . A number that terminates is 0.75. In the first case, 0.333... = 1/3, and in the second case 0.75 = 3/4. Both of these numbers are rational numbers because they can be expressed as the ratio between two integers. Your daughter was correct in separating the real numbers into two mutually exclusive sets: the set of rational numbers and the set of irrational numbers. These two sets of numbers have no numbers in common: that is, a number is either rational or irrational. Each set is also infinite: that is, you can't count the number of numbers within either set. Real Numbers / \ Irrational No. Rational Numbers A Venn diagram is a good way to "see" the relations among different sets of numbers: The set of rational numbers includes the integers, the whole numbers, and the counting (natural) numbers, in that order. Therefore, any number that is a natural number is also a whole number, an integer, and a rational number. Thus, starting from the bottom of the tree and proceding up, the number 3 is a natural number, a whole number, an integer, and a rational number. However, the number 0 is not a natural number - but it is a whole number, an integer, and a rational number. The number 0.5 is rational only (it terminates). It is not an integer, a whole number, or a natural number. The list your daughter was given contains the names of sets (such as the integers and the whole numbers) as well as names of elements that belong to one or more sets (for example, 0 and transcendental numbers). Therefore, to complete this classification tree, first make the tree, starting as shown in the text above. Then find examples of specific types of numbers and place them in their corresponding sets as shown in the Venn diagram. Your daughter is on the right track. Keep going. -Doctor Chita, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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