introduction
TRANSCRIPT
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MATHEMATICS-1Lecturer#1
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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
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AIMS
Reinforcement of basic numeracy and algebraic manipulation.
A combination of lectures, seminars and tutorials is used to explain concepts and apply them through exercises
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Study Hours
Lectures: 48.00 Directed Study: 138.00
Seminars/Tutorials: 32.00 Formal Exams: 2.00
Laboratory/Practical: 0.00 Other: 0.00 Total: 200
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1 Assessment Type Duration (hours) Percentage
Classroom test - 25%
Description
2 classroom tests each lasting 1 hour
2 Assessment Type Duration (hours) Percentage
Examination - open book or seen paper 2 50%
Description
Examination
3 Assessment Type Duration (hours) Percentage
Coursework - 25%
Description
2 assignments consisting of Maths questions taking approx 2 hours to answer per assignment
900 Assessment Type Duration (hours) Percentage
Examination - open book or seen paper 2 100%
Description
Supplementary examination
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NUMBERS
Number is a mathematical concept used to describe and access quantity.
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Here is an interesting and lovely way to look at the beauty of mathematics, and of God, the sum of all wonders.
The Beauty of Mathematics
Wonderful World
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1 x 8 + 1 = 912 x 8 + 2 = 98
123 x 8 + 3 = 9871234 x 8 + 4 = 9876
12345 x 8 + 5 = 987 65123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 987654312345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321
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1 x 9 + 2 = 1112 x 9 + 3 = 111
123 x 9 + 4 = 11111234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 1111111112345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111
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9 x 9 + 7 = 8898 x 9 + 6 = 888
987 x 9 + 5 = 88889876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 8888888898765432 x 9 + 0 = 888888888
Brilliant, isn’t it?
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1 x 1 = 111 x 11 = 121
111 x 111 = 123211111 x 1111 = 1234321
11111 x 11111 = 123454321111111 x 111111 = 12345654321
1111111 x 1111111 = 123456765432111111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321
And look at this symmetry:
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NUMBER REPRESENTATIONThe number system 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 (÷)
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POWERS
Repeated multiplication by the 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.
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PLACE VALUE Once a number 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.
Millions Hundreds thousands
Ten thousands
Thousands Hundreds Tens units
4 0 8 7 0 2 6
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REAL NUMBERS
Real Numbers are any number on a number line. It is the combined set of the rational and irrational numbers.
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RATIONAL NUMBERS
Rational Numbers are numbers that can be expressed as a fraction or ratio of two integers.
Example: 3/5, 1/3, -4/3, -25
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IRRATIONAL NUMBERSIrrational Numbers are 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…..
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RATIO/QUOTIENT
A comparison of two 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.
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WHOLE NUMBERS
Whole numbers are 0 and all positive numbers such as 1, 2, 3, 4 ………
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INTEGERS
Any positive or negative whole numbers including zero. Integers are not decimal numbers are fractions.
. . .-3, -2, -1, 0, 1, 2, 3, …
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The Real Number System
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Real Numbers
Rational Numbers Irrational Numbers
3
1/2-2
15%
2/3
1.456
-0.7
0
3 2
-5 2
34
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The Real Number System
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Real Numbers
Rational Numbers Irrational Numbers
31/2 -2
15%
2/3
1.456- 0.7
0
3 2
-5 2
34
Integers
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The Real Number System
04/12/2023 jwaid 23
Real Numbers
Rational Numbers Irrational Numbers
31/2
-2
15%
2/3
1.456- 0.7
0
3 2
-5 2
34
Integers
Whole
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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, and
x
0 1 2 3 4 5-5 -4 -2 -1-3
2
12 2
every point on the number line represents one real number.
Properties of Real Numbers
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Real numbers can be classified as either _______ or ________.rational irrational
Rational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b
a
The decimal form of a rational number is either a terminating or repeating decimal.
Examples: ratio form decimal form
9 0.3
83
375.0
73
428571.0
or . . . 714285714285714285.0
Properties of Real Numbers
zero
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Real numbers can be classified a either _______ or ________.rational irrational
A real number that is not rational is irrational.
The decimal form of an irrational number neither __________ nor ________.terminates repeats
Examples:
. . . 141592654.3 More Digits of PI?
e . . . 718281828.2
2 3 5 7 11 13
Do you notice a pattern within this group of numbers?
They’re all PRIME numbers!
Properties of Real Numbers
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Example 1
Classify each number as being real, rational, irrational, integer, whole, and/or natural numbers. Pick all that apply.
7
12 0
10.333
6
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The square root of any whole number is either whole or irrational.
x
0 1 32 4 5 6 7 98 10
For example, is a whole number, but , since it lies between 5 and 6, must be irrational.
36 30
36
. . . 477225575.5
25
30
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,
169644936251694 14412110081
Properties of Real Numbers
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Any ?