Discrete Mathematics I Lectures Chapter 4
... Therefore x is not prime (no matter what even integer you pick!) If the negation is TRUE, then the original statement must be FALSE!!! ...
... Therefore x is not prime (no matter what even integer you pick!) If the negation is TRUE, then the original statement must be FALSE!!! ...
2. Exponents
... Suppose you have (x5)(x6); how do you simplify that? Just remember that you’re counting factors. x5 = (x)(x)(x)(x)(x) and x6 = (x)(x)(x)(x)(x)(x) Now multiply them together: (x5)(x6) = (x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) = x11 Why x11? Well, how many x’s are there? Five x factors from x5, and six x fa ...
... Suppose you have (x5)(x6); how do you simplify that? Just remember that you’re counting factors. x5 = (x)(x)(x)(x)(x) and x6 = (x)(x)(x)(x)(x)(x) Now multiply them together: (x5)(x6) = (x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) = x11 Why x11? Well, how many x’s are there? Five x factors from x5, and six x fa ...
A Brief History of Pi
... A Brief History of Pi Ancient civilizations knew that there was a fixed ratio of circumference to diameter that was approximately equal to three. The Greeks refined the process and Archimedes is credited with the first theoretical calculation of Pi. In 1761 Lambert proved that Pi was irrational, tha ...
... A Brief History of Pi Ancient civilizations knew that there was a fixed ratio of circumference to diameter that was approximately equal to three. The Greeks refined the process and Archimedes is credited with the first theoretical calculation of Pi. In 1761 Lambert proved that Pi was irrational, tha ...
Dice Probabilities, Intro to Binomial Probabilities
... e) Notice that the probability of getting exactly 4 heads when 7 coins are tossed is the same as the probability of getting another number of heads. What is that number? f) The probability of getting exactly 2 heads when 10 coins are tossed is the same as getting another number of heads. What is tha ...
... e) Notice that the probability of getting exactly 4 heads when 7 coins are tossed is the same as the probability of getting another number of heads. What is that number? f) The probability of getting exactly 2 heads when 10 coins are tossed is the same as getting another number of heads. What is tha ...
7.5 Descartes` Rule of Signs
... determine the number of possible positive and negative zeros. • The Upper and Lower Bound Theorem is used to determine the bounds of the zeros. • The Intermediate Value Theorem is used to determine between which real numbers a zero occurs. ...
... determine the number of possible positive and negative zeros. • The Upper and Lower Bound Theorem is used to determine the bounds of the zeros. • The Intermediate Value Theorem is used to determine between which real numbers a zero occurs. ...
Methods of Proof
... Existence: that such a value does indeed exist Either via a constructive or non-constructive existence ...
... Existence: that such a value does indeed exist Either via a constructive or non-constructive existence ...
Addition
Addition (often signified by the plus symbol ""+"") is one of the four elementary, mathematical operations of arithmetic, with the others being subtraction, multiplication and division.The addition of two whole numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two apples together; making a total of 5 apples. This observation is equivalent to the mathematical expression ""3 + 2 = 5"" i.e., ""3 add 2 is equal to 5"".Besides counting fruits, addition can also represent combining other physical objects. Using systematic generalizations, addition can also be defined on more abstract quantities, such as integers, rational numbers, real numbers and complex numbers and other abstract objects such as vectors and matrices.In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others. In algebra, addition is studied more abstractly.Addition has several important properties. It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see Summation). Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months and even some non-human animals. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.