Rational and Irrational Numbers.doc

7/27/2019 Rational and Irrational Numbers.doc

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Examples:

Rational Numbers

OK. A Rational Number can be written as a Ratio of two integers (ie a simple fraction).

Example: 1.5 is rational, because it can be written as the ratio 3/2

Example: 7 is rational, because it can be written as the ratio 7/1

Example 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3

Irrational Numbers

But some numbers cannot be written as a ratio of two integers ...

...they are called Irrational Numbers.

It is irrationalbecause it cannot be written as a ratio (or fraction),

not because it is crazy!

Example: (Pi) is a famous irrational number.

= 3.1415926535897932384626433832795 (and more...)

You cannot write down a simple fraction that equals Pi.

The popular approximation of22/7 = 3.1428571428571... is close but not accurate.

Another clue is that the decimal goes on forever without repeating.

Rational vs Irrational

So you can tell if it is Rational or Irrational by trying to write the number as a simple fraction.

Example: 9.5 can be written as a simple fraction like this:

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9.5 = 19/2

So it is a rational number (and so is not irrational)

Here are some more examples:

Number As a Fraction Rational orIrrational?

1.75 7/4 Rational

.001 1/1000 Rational

2

(square root of 2) ?Irrational !

Square Root of 2

Let's look at the square root of 2 more closely.

If you draw a square (of size "1"),

what is the distance across the

diagonal?

The answer is the square rootof 2, which is 1.4142135623730950...(etc)

But it is not a number like 3, or five-thirds, or anything like that ...

... in fact you cannot write the square root of 2 using a ratio of two numbers

... I explain why on theIs It Irrational? page,

... and so we know it is an irrational number

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Famous Irrational Numbers

Pi is a famous irrational number. People have calculated Pi to over one

million decimal places and still there is no pattern. The first few digits look

like this:

3.1415926535897932384626433832795 (and more ...)

The numbere (Euler's Number) is another famous irrational number. People

have also calculated e to lots of decimal places without any pattern showing.

The first few digits look like this:

2.7182818284590452353602874713527 (and more ...)

TheGolden Ratiois an irrational number. The first few digits look like this:

1.61803398874989484820... (and more ...)

Many square roots, cube roots, etc are also irrational numbers. Examples:

3 1.7320508075688772935274463415059 (etc)

9

9

9.9498743710661995473447982

100121 (etc)

But 4 = 2 (rational), and 9 = 3 (rational) ...

... so not all roots are irrational.

Note on Multiplying Irrational Numbers

Have a look at this:

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= 2 is irrational But 2 2 = 2 is rational

So be careful ... multiplying irrational numbers can result in a rational number!

History of Irrational Numbers

Apparently Hippasus (one ofPythagoras'students) discovered irrational numbers when trying to represent the

square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write the square root

of 2 as a fraction and so it was irrational.

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers

had perfect values. But he could not disprove Hippasus'"irrational numbers" and so Hippasus was thrown

overboard and drowned!

Surds

If you can't simplify a number to remove a square root (or cube root etc) then it is a surd.

Example: 2 (square root of 2) can't be simplified further so it is a surd

Example: 4 (square root of 4) can be simplified (to 2), so it is not a surd!

Have a look at some more examples:

Numb

er

Simplif

ed

As a

Decimal

Surd or

not?

2 21.4142135...

(etc)Surd

3 31.7320508...

(etc)Surd

4 2 2 Not a surd

(1/4) 1/2 0.5 Not a surd

3(11) 3(11)2.2239800...

(etc)Surd

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3(27) 3 3 Not a surd

5(3) 5(3)1.2457309...

(etc)Surd

As you can see, the surds have a decimal which goes on forever without repeating, and are actuallyIrrational

Numbers.

In fact "Surd" used to be another name for "Irrational", but it is now used for

a root that is irrational.

How did we get the word "Surd" ?

Well around 820 AD al-Khwarizmi(the Persian guy who we get the name

"Algorithm" from) called irrational numbers "'inaudible" ... this was later translated

to the Latinsurdus ("deaf" or "mute")

Conclusion

If it is a root and irrational, it is a surd.

But not all roots are surds.

Squares and Square Roots

First learn about Squares, then Square Roots are easy.

How to Square A Number

To square a number, just multiply it by itself ...

Example: What is 3 squared?

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3 Squared = = 3 3 = 9

"Squared" is often written as a little 2 like this:

This says "4 Squared equals 16"

(the little 2 says the number appears twice in multiplying)

Squares From12to621 Squared = 12 = 1 1 = 1

2 Squared = 22 = 2 2 = 4

3 Squared = 32 = 3 3 = 9

4 Squared = 42 = 4 4 = 16

5 Squared = 52 = 5 5 = 25

6 Squared = 62 = 6 6 = 36

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You can also find the squares on

theMultiplication Table:

Negative Numbers

You can also square negative numbers.

Example: What happens when you square (-5) ?Answer:

(-5) (-5) = 25

(because a negative times a negative gives a positive)

That was interesting!

When you square a negative number you get a positive result.

Just the same as if you had squared a positive number:

(For more detail read Squares and Square Roots in Algebra)

Note: if someone says "minus 5 squared" do you:

Square the 5, then do the minus?

Or do you square (-5) ?

You get different answers:

Square 5, then do the

minus:Square (-5):

-(55) = -25(-5)(-5)

= +25

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Always make it clear what you mean, and that is what the "( )" are for.

Square Roots

A square root goes the other way:

3 squared is 9, so a square root of 9 is 3

A square root of a number is ...

... a value that can be multiplied by itselfto give the original number.

A square root of9 is ...

... 3, because when 3 is multiplied by itselfyou get 9.

It is like asking:

What can I multiply by itself to get this?

To help you remember think of the root of a tree:

"I know the tree, but what is the root that produced it?"

In this case the tree is "9", and the root is "3".

Here are some more squares and square roots:

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4 16

5 25

6 36

Decimal Numbers

You can also square decimal numbers.

Try the sliders below. Note: the numbers here are only shown to 2 decimal places.

View

Larger

Using the sliders (remembering it is only accurate to 2 decimal places):

What is the square root of8? What is the square root of9?

What is the square root of10?

What is 1 squared?

What is 1.1 squared?

What is 2.6 squared?

The Square Root Symbol

This is the special symbol that means "square root", it is sort of like a tick, and actually

started hundreds of years ago as a dot with a flick upwards.

It is called the radical, and always makes math look important!

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You can use it like this:

you would say "square root of 9 equals 3"

Example: What is 25?

Well, we just happen to know that 25 = 5 5, so if you multiply 5 by itself (5 5) you will get 25.

So the answer is:

25 = 5Example: What is 36 ?

Answer: 6 6 = 36, so 36 = 6

Perfect Squares

The perfect squares are the squares of the whole numbers:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 etc

Perfect

Squares:1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 ...

Try to remember at least the first 10 of those.

Calculating Square Roots

It is easy to work out the square root of a perfect square, but it is really hard to work out other square roots.

Example: what is 10?

Well, 3 3 = 9 and 4 4 = 16, so we can guess the answer is between 3 and 4.

Let's try 3.5: 3.5 3.5 = 12.25

Let's try 3.2: 3.2 3.2 = 10.24

Let's try 3.1: 3.1 3.1 = 9.61

...

Getting closer to 10, but it will take a long time to get a good answer!

At this point, I get out my calculator and it says:

3.1622776601683793319988935444327

But the digits just go on and on, without any pattern.

So even the calculator's answer is only an approximation !

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Note: numbers like that are calledIrrational Numbers, if you want to know more.

A Fun Way to Calculate a Square Root

There is a fun method for calculating a square root that gets more and more accurate each time around:

a) start with a guess (let's guess 4 is the square root of 10)

b) divide by the guess (10/4 = 2.5)

c) add that to the guess (4 + 2.5 = 6.5)

d) then divide thatresult by 2, in other words halve it. (6.5/2 = 3.25)

e) now, set that as the new guess, and start at b) again

Our first attempt got us from 4 to 3.25

Going again (b to e) gets us: 3.163

Going again (b to e) gets us: 3.1623

And so, after 3 times around the answer is 3.1623, which is pretty good, because:

3.1623 x 3.1623 = 10.00014

Now ... why don't you try calculating the square root of 2 this way?

How to Guess

What if you have to guess the square root for a difficult number such as "82,163" ... ?

In that case I would think to myself "82,163" has 5 digits, so the square root might have 3 digits

(100x100=10,000), and the square root of 8 (the first digit) is about 3 (3x3=9), so 300 would be a good start.

Cubes and Cube Roots

To understand cube roots, first you must understand cubes ...

How to Cube A Number

To cube a number, just use it in a multiplication 3 times ...

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Example: What is 3 Cubed?

3 Cubed = = 3 3 3 = 27

Note: we write down "3 Cubed" as 33

(the little "3" means the number appears three times in multiplying)

Some More Cubes

4 cubed = 43 = 4 4 4 = 64

5 cubed = 53 = 5 5 5 = 125

6 cubed = 63 = 6 6 6 = 216

Cube Root

A cube root goes the other direction:

3 cubed is 27, so the cube root of 27 is 3

3 27

The cube root of a number is ...

... a special value that when cubed gives the original number.

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The cube root of27 is ...

... 3, because when 3 is cubed you get 27.

Note: When you see "root" think

"I know the tree, but what is the root that produced it?"

In this case the tree is "27", and the cube root is "3".

Here are some more cubes and cube roots:

4 64

5 125

6 216

Example: What is the Cube root of 125?

Well, we just happen to know that 125 = 5 5 5 (if you use 5 three times in a multiplication you will get 125) ...

... so the answer is 5

The Cube Root Symbol

This is the special symbol that means "cube root", it is the "radical"symbol (used for

square roots) with a little three to mean cube root.

You can use it like this: (you would say "the cube root of 27 equals 3")

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You Can Also Cube Negative Numbers

Have a look at this:

If you cube 5 you get 125: 5 5 5 = 125

If you cube -5 you get -125: -5 -5 -5 = -125

So the cube root of -125 is -5

Perfect Cubes

The Perfect Cubes are the cubes of thewhole numbers:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 etc

Perfect

Cubes:1 8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 ...

It is easy to work out the cube root of a perfect cube, but it is really hard to work out other cube roots.

Example: what is the cube root of 30?

Well, 3 3 3 = 27 and 4 4 4 = 64, so we can guess the answer is between 3 and 4.

Let's try 3.5: 3.5 3.5 3.5 = 42.875 Let's try 3.2: 3.2 3.2 3.2 = 32.768

Let's try 3.1: 3.1 3.1 3.1 = 29.791

We are getting closer, but very slowly ... at this point, I get out my calculator and it says:

3.1072325059538588668776624275224

... but the digits just go on and on, without any pattern. So even the calculator's answer is only

an approximation !

(Further reading: these kind of numbers are calledsurdswhich are a special type ofirrational number)

Simplifying Square Roots

To simplify a square root: make the number inside the square root as small as possible (but still a whole number):

Example: 8 is simpler as 22

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Get your calculator and check if you want: they are both the same value!

This is the useful rule to remember:

And this is how to use it:

Example (continued)

8 = (42) = 4 2 = 22

(Because the square root of 4 is 2)

Here is another example:

Example: simplify 12

12 is 4 times 3:

12 = (4 3)

Use the rule:

(4 3) = 4 3

And the square root of 4 is 2:

4 3 = 23

So 12 is simpler as 23

And here is how to simplify in one line:

Example: simplify 18

18 = (9 2) = 9 2 = 32

It often helps tofactor the numbers (into prime numbersis best):

Example: simplify 6 15

First we can combine the two numbers:

6 15 = (6 15)

Then we factor them:

(6 15) = (2 3 3 5)

Then we see two 3s, and decide to "pull them out":

(2 3 3 5) = (3 3) (2 5) = 310

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Fractions

There is a similar rule for fractions:

Example: simplify 30 / 10

First we can combine the two numbers:

30 / 10 = (30 / 10)

Then simplify:

(30 / 10) = 3

A Harder Example

Example: simplify (20 5) / 2

See if you can follow the steps:

(20 5)/2

((2 2 5) 5)/2

(2 2 5 5)/2

2 5 5

2 5

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Surds

Note: the root that you can't simplify further is called a Surd.

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Rational and Irrational Numbers.doc