September 11, 2012
Rational and Irrational Numbers
Rational and irrational are words that you hear outside of mathematics.
Discuss with a partner what you think when I say the words rational and
irrational. Be prepared to share with the class.
A rational person:
· Makes good choices
· Thinks logically
· Is organized
· Weighs the pros and cons
of a situation
· ???
An irrational person:
·
·
·
·
·
Is random
Makes poor choices
doesn't think logically
Is guided by emotion
???
September 11, 2012
Rational and Irrational Numbers
Rational and irrational are words that you hear outside of mathematics.
Discuss with a partner what you think when I say the words rational and
irrational. Be prepared to share with the class.
A rational person:
· Makes good choices
· Thinks logically
· Is organized
· Weighs the pros and cons
of a situation
· ???
An irrational person:
·
·
·
·
·
Is random
Makes poor choices
doesn't think logically
Is guided by emotion
???
September 11, 2012
Rational Numbers
Irrational Numbers
In mathematics, a rational number
is a number that can be written as
the quotient of two integers. That's
just a fancy way of saying that a
rational number is a number that can
be written as a fraction.
In mathematics, an irrational
number is a number that cannot
be written as a quotient of two
integers. Again, that's just a fancy
way of sating that an irrational
number is a number that can't be
written as a fraction.
· A rational number can be an integer
(whole number)
· Irrational numbers can be special
· A rational number can be a fraction
numbers (Like pi)
· A rational number can be a decimal, · Irrational numbers can be
but the decimal has to terminate or
decimals that go on infinitely
repeat itself.
without repeating.
· Irrational numbers can be non
Examples
perfect square roots.
5, 1.25, 6 million, 4.12435, √81, 56,
Examples
pi, e, 1.5467892341568..., √2,
2
5 43 5
3
6 23 100
√57, √20, 0.0004367827687234...
September 11, 2012
So now we know what a square root is. How do we go about finding
them?
As we learned before, there are two types of square roots:
perfect squares (√4, √9, √81, etc)
non-perfect squares (√2, √23, √99, etc.)
There are also two types of numbers:
Rational Numbers are numbers that can be expressed as the
quotient of two integers. That's just a fancy way of saying that rational
numbers are numbers that can be written as fractions.
5, 2/3, 5.24, 27/5
are all rational numbers because they can be written as fractions.
Irrational numbers are numbers that can not be written as fractions.
At this point, there is only one irrational number that you know: pi
Pi is irrational because it is a decimal that goes on infinitely without
repeating and because it cannot be written as a fraction.
Non-perfect squares are also irrational numbers because if we try to
compute them, we get decimals that go on forever without repeating, just
like pi.
September 11, 2012
So now we know what a square root is. How do we go about finding
them?
As we learned before, there are two types of square roots:
perfect squares (√4, √9, √81, etc)
non-perfect squares (√2, √23, √99, etc.)
There are also two types of numbers:
Rational Numbers are numbers that can be expressed as the
quotient of two integers. That's just a fancy way of saying that rational
numbers are numbers that can be written as fractions.
5, 2/3, 5.24, 27/5
are all rational numbers because they can be written as fractions.
Irrational numbers are numbers that can not be written as fractions.
At this point, there is only one irrational number that you know: pi
Pi is irrational because it is a decimal that goes on infinitely without
repeating and because it cannot be written as a fraction.
Non-perfect squares are also irrational numbers because if we try to
compute them, we get decimals that go on forever without repeating, just
like pi.
September 11, 2012
Let's practice that:
For each group of numbers, write an R if the whole group is rational, an I
if the whole group is irrational, and a B if the group is made up of both
rational and irrational numbers.
1. 2, 5, 2/3, 7, 12
2. √4, √49, √225, √100, √196
3. √81, √389, √2, 5, √7
4. pi, 5, 2.435, 9, √12
5. √2, √5, √13, √27, √54
6. √25, 23/5, 19/4, √5, 6.43567
7. 1.33333, √12, √15, √9, √1
8. 1, 2, 3, 4, 5, 6
9. 2/3, 3/4, √16, √36, 15
10. pi, √500, √150, √320, √240
R 1. 2, 5, 2/3, 7, 12
R 2. √4, √49, √225, √100, √196
B 3. √81, √389, √2, 5, √7
B 4. pi, 5, 2.435, 9, √12
I 5. √2, √5, √13, √27, √54
B 6. √25, 23/5, 19/4, √5, 6.43567
B 7. 1.33333, √12, √15, √9, √1
R 8. 1, 2, 3, 4, 5, 6
R 9. 2/3, 3/4, √16, √36, 15
I 10. pi, √500, √150, √320, √240
September 11, 2012
So how do I find a square root?
Well the perfect squares should be easy because you are memorizing the
first 25 of them:
√1 = 1
√4 = 2
√9 = 3
√16 = 4
√25 = 5
√36 = 6
√49 = 7
√64 = 8
√81 = 9
√100 = 10
√121 = 11
√144 = 12
√169 = 13
√196 = 14
√225 = 15
√256 = 16
√289 = 17
√324 = 18
√361 = 19
√400 = 20
√441 = 21
√484 = 22
√529 = 23
√576 = 24
√625 = 25
September 11, 2012
But what about those pesky non-perfect squares?
Well, I hate to say it but this is one of those cases where there is not an
easy way. Consider the following root:
√5
Well, I know it is somewhere between 2 and 3. How about 2.5?
2.5 x 2.5 = 6.25
So that's way too high! What about 2.4?
2.4 x 2.4 = 5.76
Closer, but still a little high. How about 2.3?
2.3 x 2.3 = 5.29
So this is getting a little tedious... What about 2.2?
2.2 x 2.2 = 4.84
AHHH! Now I'm too low! Surely 2.25 will do the trick!
2.25 x 2.25 = 5.0625
√5 ∼ 2.25
That's about as close as we're going to be able to get. Remember, nonperfect squares are irrational. that means they are decimals that go on
forever, If we tried to be exact, we would have to sit her for the rest of our
lives trying to find the square root of 5, and that would be a sad, sad life.
September 11, 2012
Luckily, you're rarely going to be asked to approximate a non-perfect
square root without a calculator, so there is a trick we can employ in
order to estimate the values:
Consider the following root:
√34
Is it a perfect square?
It's not a perfect square so we can estimate it's value by thinking about
the two perfect squares that are close to it:
What is the perfect square that comes before 34? After 34?
√25 < √34 < √36
I know the square root of 34 is somewhere in between the square roots
of 25 and 36. So the root has to be between 5 and 6! Which one is it
closer to?
√34 is closer to √36 than to √25, so I know the decimal has to be high. If I
were to guess, I'd say √34 is about 5.8!
And that's really about as close as I need to get!
September 11, 2012
Let's do another together:
√83
I know that:
√81 < √83 < √100
So...
√83 has to be between 9 and 10
83 is closer to 81 than it is to 100, so I know the decimal is going to be
low in this case. If I were to estimate I would probably say:
√83 = 9.1
As long as we are just estimating, it's not too hard is it? Do you see why
it's so important to memorize the perfect squares?
September 11, 2012
For each square root, write the two numbers the root is between, and
then approximate the decimal equivalent.
1. √13
2. √27
3. √45
4. √50
5. √72
6. √81
7. √123
8. √150
9. √200
10. √325
11. √17
12. √56
13. √24
14. √33
15√259
16. √347
17.√504
18. √600
September 11, 2012
Cube Roots: I didn't have time to make this slide, so
you do it; know your cubes through 103 (be glad it is
not to 253)!
13 = 1
23 = 8...
Be able to undo the cubing as well: ∛8 = 2...