Paradoxes in Logic, Mathematics and Computer Science
... These names are justified by the following: Fact: 1 < 0 < 1 Proof: Since 0 0, ({0},{}) ({},{}) = 0 is not true. Thus, ({},{}) < ({0},{}), i.e. 0 < 1. Also, ({},{}) ({},{0}) is not true. Thus, ({},{0}) < ({},{}), i.e. 0 < 1. However, ({0},{0}) is NOT a surreal number, since it is not true ...
... These names are justified by the following: Fact: 1 < 0 < 1 Proof: Since 0 0, ({0},{}) ({},{}) = 0 is not true. Thus, ({},{}) < ({0},{}), i.e. 0 < 1. Also, ({},{}) ({},{0}) is not true. Thus, ({},{0}) < ({},{}), i.e. 0 < 1. However, ({0},{0}) is NOT a surreal number, since it is not true ...
Odd or Even - Standards Toolkit
... Even or Odd? assessment sheets Directions (for teacher to administer assessment task): Teacher may choose whether to assess students in small groups, or as a whole class. Hand out Even or Odd? assessment sheets (one per student) Tell students “You will be identifying numbers as either even or ...
... Even or Odd? assessment sheets Directions (for teacher to administer assessment task): Teacher may choose whether to assess students in small groups, or as a whole class. Hand out Even or Odd? assessment sheets (one per student) Tell students “You will be identifying numbers as either even or ...
Common Core Learning Standards
... Four friends share three candy bars. What part of one candy bar does each friend get? Draw a model to illustrate. Renee is making pasta. She makes 3 cups of noodles using 6 cups of water. How many cups of water were needed for each cup of ...
... Four friends share three candy bars. What part of one candy bar does each friend get? Draw a model to illustrate. Renee is making pasta. She makes 3 cups of noodles using 6 cups of water. How many cups of water were needed for each cup of ...
EQUIVALENCE RELATIONS Recall that "a equiv_n b" means n | b
... 3Z: {... -6 -3 0 3 6 ...} -- multiples of 3, aka integers equiv to 0 mod 3 1+3Z: {... -5 -2 1 4 7 ...} -- one more than multiples of 3, aka integers equiv to 1 mod 3 2+3Z: {... -4 -1 2 5 8 ...} -- two more than multiples of 3, aka integers equiv to 2 mod 3 This partition of Z is { 3Z, 1+3Z, 2+3Z }. ...
... 3Z: {... -6 -3 0 3 6 ...} -- multiples of 3, aka integers equiv to 0 mod 3 1+3Z: {... -5 -2 1 4 7 ...} -- one more than multiples of 3, aka integers equiv to 1 mod 3 2+3Z: {... -4 -1 2 5 8 ...} -- two more than multiples of 3, aka integers equiv to 2 mod 3 This partition of Z is { 3Z, 1+3Z, 2+3Z }. ...
Introduction to Significant Figures & Scientific Notation
... Large Numbers • If the number you start with is greater than 1, the exponent will be positive • Write the number 39923 in scientific notation • First move the decimal until 1 number is in front – 3.9923 • Now at x 10 – 3.9923 x 10 • Now count the number of decimal places that you moved (4) • Since ...
... Large Numbers • If the number you start with is greater than 1, the exponent will be positive • Write the number 39923 in scientific notation • First move the decimal until 1 number is in front – 3.9923 • Now at x 10 – 3.9923 x 10 • Now count the number of decimal places that you moved (4) • Since ...
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.