Slide 1
... These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th cen ...
... These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th cen ...
Section 3.2: Direct Proof and Counterexample 2
... Section 3.2: Direct Proof and Counterexample 2 In the last section, we considered properties of integers and proved and disproved many different facts about integers. In this section, we shall consider rational numbers. First, we shall formally define a rational number, the we shall prove some eleme ...
... Section 3.2: Direct Proof and Counterexample 2 In the last section, we considered properties of integers and proved and disproved many different facts about integers. In this section, we shall consider rational numbers. First, we shall formally define a rational number, the we shall prove some eleme ...
Solutions to exercises 1 File
... {1, 2, 3}, and then choosing three from {4, 5, 6, 7}, making 2 . 3 = 12 choices so far; and then order the five numbers in 5! = 120 ways, so that the final answer is 12.120 = 1440. ...
... {1, 2, 3}, and then choosing three from {4, 5, 6, 7}, making 2 . 3 = 12 choices so far; and then order the five numbers in 5! = 120 ways, so that the final answer is 12.120 = 1440. ...
3 Three III
... an arbitrary angle into 3 equal parts using just a straightedge and compass. A closed polygon with 3 sides is called a triangle. That’s what Euclid called it, a figure with three angles. Why do we all say that it is a figure with three sides? The word is 3 dimensional, or so it appears. The third Pr ...
... an arbitrary angle into 3 equal parts using just a straightedge and compass. A closed polygon with 3 sides is called a triangle. That’s what Euclid called it, a figure with three angles. Why do we all say that it is a figure with three sides? The word is 3 dimensional, or so it appears. The third Pr ...
Properties of Multiplication Associative Property of Multiplication
... Distributive Property The sum of two numbers times a third number is equal to the sum of each addend times the third number. For example: a x (b + c) = a x b + a x c ...
... Distributive Property The sum of two numbers times a third number is equal to the sum of each addend times the third number. For example: a x (b + c) = a x b + a x c ...
Multiplication and Division of Whole Numbers
... Notice that both the divisor and the quotient are factors of the dividend. To find the factors of a number, try dividing the number by 1, 2, 3, 4, 5, … Those numbers that divide into the number evenly are its factors. Continue this process until the factors start to repeat. A prime number is a natur ...
... Notice that both the divisor and the quotient are factors of the dividend. To find the factors of a number, try dividing the number by 1, 2, 3, 4, 5, … Those numbers that divide into the number evenly are its factors. Continue this process until the factors start to repeat. A prime number is a natur ...
94 Ninety-Four XCIV
... The number 94 has four divisors: 1, 2, 47, 94. The number 94 is the seventy-second deficient number: s(94) = 1 + 2 + 47 = 50 < 94. As a sum of four or fewer squares: 94 = 22 +32 +92 = 32 +62 +72 = 12 +22 +52 +82 = ...
... The number 94 has four divisors: 1, 2, 47, 94. The number 94 is the seventy-second deficient number: s(94) = 1 + 2 + 47 = 50 < 94. As a sum of four or fewer squares: 94 = 22 +32 +92 = 32 +62 +72 = 12 +22 +52 +82 = ...
Dividing with Significant Figures
... result is limited by the least accurate measurement involved in the calculation. • Division ▫ After dividing the numbers, you then round the result off so it has the same number of significant figures as the component with the smallest number of significant figures. ...
... result is limited by the least accurate measurement involved in the calculation. • Division ▫ After dividing the numbers, you then round the result off so it has the same number of significant figures as the component with the smallest number of significant figures. ...
Parity of zero
Zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. The simplest way to prove that zero is even is to check that it fits the definition of ""even"": it is an integer multiple of 2, specifically 0 × 2. As a result, zero shares all the properties that characterize even numbers: 0 is divisible by 2, 0 is neighbored on both sides by odd numbers, 0 is the sum of an integer (0) with itself, and a set of 0 objects can be split into two equal sets.Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as even − even = even, require 0 to be even. Zero is the additive identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively defined. Applications of this recursion from graph theory to computational geometry rely on zero being even. Not only is 0 divisible by 2, it is divisible by every power of 2, which is relevant to the binary numeral system used by computers. In this sense, 0 is the ""most even"" number of all.Among the general public, the parity of zero can be a source of confusion. In reaction time experiments, most people are slower to identify 0 as even than 2, 4, 6, or 8. Some students of mathematics—and some teachers—think that zero is odd, or both even and odd, or neither. Researchers in mathematics education propose that these misconceptions can become learning opportunities. Studying equalities like 0 × 2 = 0 can address students' doubts about calling 0 a number and using it in arithmetic. Class discussions can lead students to appreciate the basic principles of mathematical reasoning, such as the importance of definitions. Evaluating the parity of this exceptional number is an early example of a pervasive theme in mathematics: the abstraction of a familiar concept to an unfamiliar setting.