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REAL NUMBERS
AND THE NUMBER LINE
Math 230 Presentation
By Sigrid Robiso
Rational Numbers
• The ancient Greeks
created a reasonable
method of measuring
• What is a rational
number?
• Is every number a
rational number?
Euclid
Pythagorean Theorem
Pythagoras
• Remember the Pythagorean Theorem:
a² + b² = c²
Pythagorean Theorem
a² + b² = c²
example if a= 1 and b= 1
1² + 1² = c²
c= √ 2
which is an irrational number because it is not
equal to the ratio of two numbers
The Real Number Line
The Real Number Line
• Any rational number corresponds to any point on
this line
• For ex. 5/2 on the line
• But how do we find an irrational
point on the number line?
Irrational Numbers on the Number Line
1. Build a square whose base is the interval from 0 to 1
Irrational Numbers on the Number Line
2. Next draw the diagonal from 0 to the upper right corner
of the square
3. Using a compass copy the length of that diagonal line
onto the number line and make a mark
Irrational Numbers on the Number Line
• Remember: √ 2 is irrational
which makes us question is there a
uniform method to label every point on
the line- rational and irrational?
The Decimal Point
• Let’s consider the decimal expansion of √ 2
√ 2 = 1.414213562…
√ 2= 1.414213562
•
The number left to the
decimal point shows
that our number will be
somewhere between 1
and 2
•
Where? We cut the
interval from 1 to 2
into 10 equal pieces
•
The next digit, 4, tells
us in which small
interval our number is
located. We then take
that small interval and
cut it up into 10 very
small equal pieces
•
The next digit, 1, tells
us which very small
interval our number
resides
The Decimal Point
√ 2= 1.414213562….
• Notice: as we continue this
process we break the intervals
smaller and smaller
• This process allows us to pin point
our number √ 2
• We keep getting closer to the √ 2
but this process never ends for √ 2
because it is irrational
• We keep pin pointing into smaller
and smaller intervals but we must
repeat this process infinitely many
times to pinpoint the placement of
√ 2 exactly