treating the essence of problems having only one solution‚ fraction or integer. Congruence relation Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations of the ring of integers: addition‚ subtraction‚ and multiplication. For a positive integer n‚ two integers a and b are said to be congruent modulo n‚ written: if their difference a − b is an integer multiple of n (or n divides a − b). The number n is called the
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form a new set of numbers called Whole number denoted by W -1‚-2‚-3……………..- are the negative of natural numbers. The negative of natural numbers‚ 0 and the natural number together constitutes integers denoted by Z. The numbers which can be represented in the form of p/q where q 0 and p and q are integers are called Rational numbers. Rational numbers are denoted by Q. If p and q are coprime then the rational number is in its simplest form. Irrational numbers are the numbers which are non-terminating
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Chapter Summary #1 (Adv. Algebra) Real numbers. The natural numbers‚ whole numbers‚ integers numbers‚ rational numbers and irrational numbers are all subsets of the real numbers. Each real number corresponds to a point on the number line. A real numbers distance from zero on the number line its absolute value. -Natural Numbers. ( ) Natural numbers are the numbers used for counting. Example: 1‚ 2‚ 3‚ 4‚ 5… -Negative Numbers. ( ) Then man thought about numbers between
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statement is true or false a) Every counting number is an integer b) Zero is a counting number c) Negative six is greater than negative three d) Some of the integers is natural numbers 2. List the number describe and graph them on the number line a) The counting number smaller than 6 b) The integer between -3 and 3 3. Given S = {-3‚ 0‚[pic]‚ [pic]‚ e‚ ‚ 4‚ 8…}‚ identify the set of (a) natural numbers (b) whole numbers (c) integers (d) rational numbers (e) irrational numbers
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LEADING. and GRE are registered trademarks of Educational Testing Service (ETS). Table of Contents ARITHMETIC .............................................................................................................................. 1 1.1 Integers.................................................................................................................................. 1 1.2 Fractions ...................................................................................................
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LEADING. and GRE are registered trademarks of Educational Testing Service (ETS). Table of Contents ARITHMETIC .............................................................................................................................. 1 1.1 Integers.................................................................................................................................. 1 1.2 Fractions ..................................................................................................
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“What is your first number”Input 1numDisplay “What is your second number”Input 2numsumNum = 1num+2numDisplay “sum num” Convert the While loop in the following code to Do-While loops: Declare Integer x = 1 While x > 0 Display “Enter a number. “ Input x End While Declare x as Integer = 1 Do Display “Enter a number.” Input x If x ! = 0 then
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then x0 = 5 -2y for every y‚ which is not possible. 8. Determine whether the following argument is valid: pr qr (p q) ________ r Ans: Not valid: p false‚ q false‚ r true 9. Prove that the following is true for all positive integers n: n is even if and only if 3n2 8 is even. Ans: If n is even‚ then n 2k. Therefore 3n2 8 3(2k)2 8 12k2 8 2(6k2 4)‚ which is even. If n is odd‚ then n 2k 1. Therefore 3n2 8 3(2k 1)2 8 12k2 12k
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Unit 2: Checklist Higher tier (43602H) recognise integers as positive or negative whole numbers‚ including zero work out the answer to a calculation given the answer to a related calculation multiply and divide integers‚ limited to 3-digit by 2-digit calculations multiply and divide decimals‚ limited to multiplying by a single digit integer‚ for example 0.6 × 3 or 0.8 ÷ 2 or 0.32 × 5 or limited to multiplying or dividing by a decimal to one significant figure‚ for example 0.84 × 0.2 or 6.5 ÷ 0
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Grade 12 Cluster Core Subject Mathematics Student Name Student Number Section Coverage SAT I‚ basic reasoning questions. 1. If 10 + x is 5 more than 10‚ what is the value of 2x? (A) −5 (B) 5 (C) 10 (D) 25 (E) 50 2. If x and y are positive integers‚ what are all the solutions (x‚ y) of the equation 3x + 2y = 11? (A) (1‚4) only (B) (3‚1) only (C) (1‚4) and (2‚2) (D) (1‚4) and (3‚1) (E) (2‚2) and (3‚1) 3. When 70‚000 is written as 7.0 × 10n‚ what is the value of n? (A) 1 (B) 2 (C) 3 (D) 4
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