Number Sense

Question 1: Explain the term binary tree.

Answer 1: A binary tree is graphical representation of a set of ordered elements that is used in discrete mathematics. In computer science, the binary tree has the practical application of a data structure. A binary tree consists of internal nodes and leaves. The nodes are like forks in a tree that have exactly two branches (hence the term binary), right and left, extending from them. The leaves are at the end of branches and form the terminal points of the tree. A binary tree is defined recursively as either being empty (represented by a leaf) or as having a root node which branches into right and left subtrees (represented by a fork). Binary trees, also known as extended binary trees, are described by the number of nodes they have. Ordered tree is a related term. It refers to a binary tree in which the order of the subtrees is important.

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Question 2: Define discrete mathematics, state some of its branches, and explain its application.

Answer 2: Discrete mathematics, also known as finite mathematics, is the branch of mathematics which studies objects whose values are separate and distinct. It contrasts with continuous mathematics which deals with objects whose values can change in infinitely small increments. Discrete objects are often characterized by countable sets, such as integers, whereas continuous objects may be characterized by the set of real numbers. Some forms of discrete mathematics are set theory, number theory, functions, congruence and recurrence relations, linear and matrix algebra, combinatorics (the study of the combination of discrete objects), statistics, probability, logic, and graph theory.Another important branch of discrete mathematics is algorithms (the study of methods of calculation); therefore, discrete mathematics is useful in computer science. For that reason, discrete mathematics has gained importance in recent decades.

Question 3: State the function which produces the Fibonacci sequence.State the first 15 terms of the Fibonacci sequence.

Answer 3: The function which produces the Fibonacci sequence is defined recursively by f(1) = 1 f(2) = 1 f(n) = f(n – 2) + f(n – 1) In other words, the first term of the sequence is 1, the second term of the sequence is 1, and each subsequent term of the sequence is the sum of the previous two terms. The first 15 terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610. The Fibonacci sequence appears often in mathematics as well as in nature.

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