Mathematics is a wonderfully complex subject'

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Mathematics is a wonderfully complex subject.

• The starting point is the subject of numbers

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The study of numbers provides the basis upon
which to build.

• Numbers are to mathematics

1,2,3,4,5,6,7,8,9....
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as letters are to "English."
A,B,C,D,E,F,G,H,I
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BUT.

• What kind of number is it?

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Is it..

• Natural
• or

• NOT
• NATURAL?

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Is it

• Rational
• or

• IRRATIONAL?

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Is it

• REAL?

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WELCOME TO

• The Sixth Grade..

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Number

• Review

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Well take another look at

• What are natural and not natural numbers?
• What are integers?
• What are rational numbers?
• What are irrational numbers?
• Lets go..

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First of all.what are Real Numbers?

• Any number that can name a position on a number
  line is a real number. Every position on a number
  line can be named by a real number in some form.
• Well use this diagram to help

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Real Numbers r
Rational Numbers
Integers
Whole Numbers
Natural Numbers
Irrational Numbers
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What are Natural Numbers?

• Natural numbers are sometimes called counting
  numbers, 1,2,3
• Zero is not a natural number
• Negative Numbers are not Natural!

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What are Whole Numbers?

• Whole numbers are the natural numbers
• Plus zero

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What are Integers?

• Integers are the whole numbers and,

Negative whole numbers
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For example,

• 789,472,235

28 , -28
and zero are integers
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It is often useful to think of the integers as
points along a 'number line'
Note that zero is neither positive nor negative.
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The terms even and odd only apply to integers

• 2.5 is neither even nor odd.
• Zero, on the other hand, is even since it is 2
  times some integer it's 2 times 0.

• To check whether a number is odd, see whether
  it's one more than some even number
• 7 is odd since it's one more than 6, which is
  even.

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Is it Rational?

• A rational number is a number that can be written
  as a ratio 12, or fraction ½
• both the numerator (the number on top) and the
  denominator (the number on the bottom) are whole
  numbers
• All integers are rational

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For example

• The number 8 is a rational number because it can
  be written as the fraction 8/1.
• Likewise, 3/4 is a rational number because it can
  be written as a fraction.
• Even a big, clunky fraction like
• 7,324,908
• 56,003,492
• is rational, simply because it can be written
  as a fraction.

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Is it Irrational?

• A number that cannot be written as a ratio (or
  fraction), is not rational. It is considered
  irrational (not because it is crazy! )
• An irrational number can be written as a decimal,
  but not as a fraction.
• An irrational number has endless non-repeating
  digits to the right of the decimal point.
• Can you think of any?

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Some Irrational Numbers

• Pi -The ratio of the Circumference to
  theDiameter of a Circle, (the symbol is the
  Greek letter p) is irrational
• p is approximately equal to
• 3.14159265358979323846
• .333333 is not irrational since it can be written
  as a fraction 1/3. It is rational.

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Many square roots, cube roots, etc. are also
Irrational Numbers

• Examples
• The square root of 2 1.414213.
• The cube root of 3 1.732050
• But the square root of 4 2, and of 9 3, so
  not all roots are irrational.

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In Summary

• Natural numbers are the numbers 1,2,3 also known
  as counting numbers
• Whole numbers are the natural numbers plus zero
• Integers are all natural numbers and their
  negative counterparts and zero 2,-1,0,1,2.
• Rational numbers can be written as fractions and
  are sometimes called real numbers.
• Irrational numbers are decimals that never
  repeat.
• All Numbers are REAL

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Lets continue and look at

• Factors
• Prime Numbers
• Composite Numbers
• and just for fun-
• Perfect Numbers

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What is a Factor?

• A number may be made by multiplying two or more
  other numbers together. The numbers that are
  multiplied together are called factors of the
  final number.
• All numbers have a factor of one since one
  multiplied by any number equals that number.
• All numbers can be divided by themselves to
  produce the number one. Therefore, we normally
  ignore one and the number itself as useful
  factors.

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What is a Prime Number?

• A prime number is a positive integer greater than
  1 that has exactly 2 positive integer factors,
  itself and 1.
• The prime numbers start with
• 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47.and go
  on forever.

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What is a Composite Number?

• A composite number is a number greater than 1
  that has more than 2 factors.
• Examples include
• 422
• 623
• 933 and,
• 12223

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What is a Perfect Number?

• A positive integer
• that is equal to the sum of its positive integral
  factors,
• including 1 but
• excluding itself.
• Can you think of any?
• (It is unknown if there are any odd perfect
  numbers.)

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The first three perfect numbers are

• 496 1 2 4 8 16 31 62 124 248
• 28
• 1 2 4 7 14, and
• 6
• 1 2 3
• Many cultures endowed these integers with special
  religious and magical significance.

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Now on to

• Greatest Common Factors (GCF)
• Least Common Multiples (LCM)
• And Prime Factorization.