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Introduction
This document provides an overview of key concepts in mathematics including: 1) It describes the aims of the Mathematics 1 module which are to reinforce basic numeracy and algebraic manipulation through lectures, seminars and tutorials. 2) It covers various topics in numbers such as place value, real numbers, rational numbers, integers, and properties of number systems. 3) Examples are provided to classify numbers as real, rational, irrational, integer, whole or natural numbers.
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Introduction
- 1.
- 2. Module Title: Mathematics 1 Module Type: Standard module Academic Year: 2010/11, Module Code: EM-0001D Module Occurrence: A, Module Credit: 20 Teaching Period: Semester 1 Level: Foundation
- 3. AimsReinforcement of basicnumeracy and algebraic manipulation.A combination of lectures, seminars and tutorials is used to explain concepts and apply them through exercises
- 4. Study HoursLectures: 48.00DirectedStudy: 138.00 Seminars/Tutorials: 32.00Formal Exams: 2.00 Laboratory/Practical: 0.00Other: 0.00Total: 200
- 6. NumbersNumber is amathematical concept used to describe and access quantity.
- 7. The Beauty ofMathematicsHere is an interesting and lovely way to look at the beauty of mathematics, and of God, the sum of all wonders.Wonderful World
- 8. 1 x 8+ 1 = 912 x 8 + 2 = 98123 x 8 + 3 = 9871234 x 8 + 4 = 987612345 x 8 + 5 = 98765123456 x 8 + 6 = 9876541234567 x 8 + 7 = 987654312345678 x 8 + 8 = 98765432123456789 x 8 + 9 = 987654321
- 9. 1 x 9+ 2 = 1112 x 9 + 3 = 111123 x 9 + 4 = 11111234 x 9 + 5 = 1111112345 x 9 + 6 = 111111123456 x 9 + 7 = 11111111234567 x 9 + 8 = 1111111112345678 x 9 + 9 = 111111111123456789 x 9 +10= 1111111111
- 10. 9 x 9+ 7 = 8898 x 9 + 6 = 888987 x 9 + 5 = 88889876 x 9 + 4 = 8888898765 x 9 + 3 = 888888987654 x 9 + 2 = 88888889876543 x 9 + 1 = 8888888898765432 x 9 + 0 = 888888888 Brilliant, isn’t it?
- 11. And look atthis symmetry:1 x 1 = 111 x 11 = 121111 x 111 = 123211111 x 1111 = 123432111111 x 11111 = 123454321111111 x 111111 = 123456543211111111 x 1111111 = 123456765432111111111 x 11111111 = 123456787654321111111111 x 111111111 = 12345678987654321
- 12. Number RepresentationThe numbersystem that we use today has taken thousand of years to develop. The Arabic system that we commonly use consists of exactly ten symbols:0 1 2 3 4 5 6 7 8 9Each symbol is called a digit. Our system involves counting in tens. This type of system is called denary system, and 10 is called the base of the system.It is possible to use a number other than 10. For example, computer systems use base 2( the binary system)Numbers are combined together, using the four arithmetic operations.addition (+), subtraction (-), multiplication (×) and division (÷)
- 13. PowersRepeated multiplication bythe same number is known as raising to a power. For example 8×8×8×8×8 is written 85 (8 to the power 5) Check your calculator for xy.
- 14. Place valueOnce anumber contains more then one digits, the idea of place value is used to tell us its worth. In number 2850 and 285, the 8 stands for something different. In 285, 8 stands for 8 ‘tens’. In 2850, the 8 stands for 8 ‘hundreds’. The following table show the names given to the first seven places.The number shown is 4087026, which is 4 million eighty-seven thousands and twenty-six.
- 15. Real NumbersReal Numbersare any number on a number line. It is the combined set of the rational and irrational numbers.
- 16. Rational NumbersRational Numbersare numbers that can be expressed as a fraction or ratio of two integers.Example: 3/5, 1/3, -4/3, -25
- 17. Irrational NumbersIrrational Numbersare numbers that cannot be written as a ratio of two integers. The decimal extensions of irrational numbers never terminate and never repeat.Example: – 3.45455455545555…..
- 18. Ratio/QuotientA comparison oftwo numbers by division. The ratio of 2 to 3 can be stated as 2 out of 3, 2 to 3, 2:3 or 2/3.
- 19. Whole numbersWhole numbersare 0 and all positive numbers such as 1, 2, 3, 4 ………
- 20. IntegersAny positive ornegative whole numbers including zero. Integers are not decimal numbers are fractions. . . .-3, -2, -1, 0, 1, 2, 3, …
- 21. The Real NumberSystem9/28/2010jwaid21Real NumbersRational NumbersIrrational Numbers1/2-22332/305 23415%-0.71.456
- 22. The Real NumberSystem9/28/2010jwaid22Real NumbersRational NumbersIrrational NumbersIntegers233-21/22/305 23415%1.456- 0.7
- 23. The Real NumberSystem9/28/2010jwaid23Real NumbersRational NumbersIrrational NumbersIntegers Whole 2331/22/305 234-215%1.456- 0.7
- 24. x-5-1-4-2-3152340Properties of RealNumbers All of the numbers that you use in everyday life are real numbers.Each real number corresponds to exactly one point on the number line, andevery point on the number line represents one real number.
- 25. Rational numberscan be expressed as a ratio , where a and b areintegers and b is not ____! Properties of Real Numbers Real numbers can be classified as either _______ or ________.rationalirrationalzeroThe decimal form of a rational number is either a terminating or repeating decimal.Examples: ratio form decimal form
- 26. Properties of RealNumbers Real numbers can be classified a either _______ or ________.rationalirrationalA real number that is not rational is irrational.repeatsThe decimal form of an irrational number neither __________ nor ________.terminatesExamples: More Digits of PI?Do you notice a pattern within this group of numbers?They’re all PRIME numbers!
- 27. Example 1Classify eachnumber as being real, rational, irrational, integer, whole, and/or natural numbers. Pick all that apply.
- 28. For example, is a whole number, but , since it lies between 5 and 6, must be irrational.236010945871xProperties of Real Numbers The square root of any whole number is either whole or irrational.Common Misconception:Do not assume that a number is irrational just because it is expressed using the square root symbol.Find its value first!Study Tip:KNOW and recognize (at least) these numbers,
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