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Real NumbersReal NumbersTypes of Real NumbersTypes of Real Numbers
Image source http://c.asstatic.com/images/sunnyrain43229-70600-real-numbers-rational-irrational-ecot-culver-math-education-ppt-powerpoint-180_135.jpg
Real Numbers✦ Two major groups -
✦ Rational
& Irrational
Image from http://2.bp.blogspot.com/_gWQaU40PH24/S4BfSLW_ONI/AAAAA
AAAFwM/t3H-Sngze0A/s320/rational.gif
Image source http://3.bp.blogspot.com/_r9OeV7enT6s/SctQz4QiPLI/AAAAAAAAApM/aZVDaddjsmc/S220/321.jpg
Rational Numbers✦ What does it mean to act rational?
Hmmm....
✦ Rational is something that makes sense.
✦ Rational numbers make sense because they are predictable.
✦ Rational numbers fall into 1 of 2 categories.
Image from http://2.bp.blogspot.com/_gWQaU40PH24/S4BfSLW_ONI/AAAAAAAAFwM/t3H-Sngze0A/s320/rational.gif
Rational Numbers - Category 1
✦ What do these numbers have in common?
✦ Notice they all stop or terminate.
✦ What about the fractions? Change them to decimals.
✦ Do they terminate now?
✦ Rational numbers are like the terminator because they terminate or end.
✦ The first category Rational Numbers can fall in is terminating numbers.
3
−27
−2.34 1
2
−8
4
123,456
Rational Numbers - Category 2
✦ What do these numbers have in common?
✦ Can you guess what the next number will be for each of these numbers?
✦ What about the fractions?
✦ Change them to decimals. Can you predict the next number?
✦ Notice they these numbers follow a predictable pattern.
✦ The second category Rational Number may fall in is following a predictable pattern.
5
3
9.232323...
−11
3
−0.1111...
7.89789
Irrational Numbers✦ What does it mean when a person
is irrational? Hmm....
✦ What do you notice about these numbers?
✦ Can you predict the next number?
✦ When do they end? (Put the square roots in a calculator to analyze their decimal equivalent.)
✦ Irrational numbers are unpredictable numbers just like an irrational person has unpredictable behavior.
✦ Irrational Numbers never end and the next number cannot be guessed because no pattern exists.
Image from http://bulk2.destructoid.com/ul/181777-IrrationalMainImage.jpg
π =3.14159...
e=2.718281828459...
2.1239257...
2
3
5
Types of Rational Numbers
✦ Natural Numbers (also called the Counting Numbers)
✦ What number do you naturally start with when you count?
✦ 1.
✦ The Natural Numbers are the numbers we count with beginning with 1.
✦ 1, 2, 3, 4, ...
✦ Wh0le Numbers - The natural numbers plus zero
✦ 0, 1, 2, 3, 4, ...
✦ Integers - All whole numbers and their opposites. In-other-words, all positive and negative numbers without decimals or fractions.
✦ ... -3, -2, -1, 0, 1, 2, 3, ...
✦ Notice some numbers fall into more than one category.
Real Numbers
5
RationalIrrational
Integers
Whole
Natural0
1
−7
−42
3
−52
0.6767...
π
e
2 =1.4142135...
5
41
3
Be sure you can do the following...
✦ Describe the difference between an irrational and rational number.
✦ Describe the characteristics of a natural number.
✦ Describe the characteristics of a whole number.
✦ Describe the characteristics of a integer.
✦ Give a example of
✦ an irrational number.
✦ a rational number and one that is not an integer.
✦ an integer and one that is not a whole number.
✦ a whole number and one that is not a natural number.
✦ a natural number.