THE REAL NUMBER SYSTEM
1
LESSON
Review
The real number system is a system that has been developing since the
beginning of time. By now you should be very familiar with the following
number sets :
Natural or counting numbers : 1; 2; 3; 4; …
Whole numbers : 0; 1; 2; 3; 4; …
Integers : … –3; –2; –1; 0 ; 1; 2; 3; …
Did you know? The three dots (…) are called ellipses and indicate that there
would be more digits to come before or after in the list.
And then along came all numbers that could be written as fractions and
because fractions are actually ratios the name ‘rational’ numbers was born.
FraCtioNs (Can you see the word ‘ratio’ hidden inside the word FRACTIONS?)
So what exactly then is a rational number?
By definition, any number of the form _a where a, b are integers and b ¹ 0 is
b
rational.
So let us look at the kind of numbers that fit into this family.
1.
All natural, whole and integer numbers are rational.
2 can be written as _2 which fits the definition of _a –10 can be written as
1
–10
_
which makes it rational
b
1
2.
All mixed fractions are rational
e.g: 1.
3_1 = _7
2
2.
3.
2
3
–103
=_
–10_
10
10
All terminating decimals are rational (they end or have a finite number of
decimal places)
25 _
=1
e.g. : 1.
0,25 = _
100
2.
4
32
504
4
= 4_
=_
4,032 = 4_
1 000
125 125
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If you type 4,032 into your calculator and hit equals – it might change
4
directly. Ask your teacher to show you if your calculator can
it to 4_
125
504
4
into _
as well.
change 4_
125
4.
125
All recurring decimals are rational (there is a repeat pattern identified)
․
e.g.: 1.
0,3 = _1
․․
3
32 _
1,32 = 1_
= 131
99
99
2.
If you type 0,33333333333 into your calculator and hit equals it converts
this recurring decimal into _13 for you. The same holds for 1,323232323232
32
being changed to 1_
.
99
However, you are required to be able to do this conversion manually.
So this is how its done.
․
Convert 0,3 to a common fraction. Show all working.
4.1
Let x = 0,33333 …
Then 10x = 3,33333 …
(Multiply by 10)
Now 9x = 3
x = _1
(Subtract top line from bottom)
(Divide both sides by 9)
․․
3
Convert 1,32 to a common fraction. Show all working.
4.2
Let x = 1,3232323232 …
100 x = 132,32323232 … (multiply by 100 because we need the part
after the decimal comma to be the same)
99x = 131
131
x=_
99
4.3
(subtract top line from bottom
(divide both sides by 99)
․․
Convert 4,0245 to a common fraction. Show all working.
Let x = 4,02454545
100 x = 402,454545
(multiply by 100 to bring the nonrepeating digits in front of the decimal)
10000 x = 40245,454545
9900 x = 39843
3 9843 _
= 27
x=_
9 900
4 1100
subtract second line from third line
the answer you get immediately on
your calculator
Of course, all rational numbers can be shown on a number line and there are
infinitely many of them.
What does this mean? Well if we look at the line between integers 1 and 2
1
a
b c 2
Then a = 1_12 = _32
b = 1_3 = _7
4
4
15
c = 1_78 = _
etc
8
And so we can always place another fraction between any two fractions, and
keep on going forever. This means that the rational numbers are densely
(tightly) packed on to the number line – but there is always room to squeeze in
at least one more. A weird thought!
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You would think that we have now mentioned all the numbers that exist,
wouldn’t you? However, some numbers are not able to be written in the form _ab ,
with a and b integers and b ≠ 0 and so they fail the rational test. We call this
family of numbers the IRRATIONAL numbers (and yes, they can be thought of as
mad!)
Example
_
Key
The screen should show
_
_ the value √2 into your calculator.
√2 = 1,414213562 but in fact √2 = 1,414213562 …
The rest of the digits are not visible/showing on your screen as there is limited
space. We have no way of knowing/predicting what the next digits are and
so here we have a case of a decimal which does not end (non – terminating)
and does not re-occur (non – recurring) and so this number is not rational,
irrational.
_
Here is a quick proof of how we know that √2 is irrational. It is a proof by
contradiction.
_
_
First we suppose that √2 is a rational number. So we can write √2 = _ab ; a, b are
integers b ≠ 0.
We also suppose that _ab is a fraction in simplest form.
_
square both sides
Now (√2 )2 = (_ab )2
2
a
2=_
b2
a = 2b2 … (1)
2
So now a2 is an even number since it is equal to 2 times something.
But 2 × 2 = 4; 3 × 3 = 9; 4 × 4 = 16; 5 × 5 = 25; 7 × 7 = 49
Can you see that if a2 is even than a itself has to be even (since (odd)2 = odd)
This means that a = 2k (can be written as a multiple of 2)
So now from (1) above
(2k)2
2=_
b2
2
4k
2 = _2
b
2b2 = 4k2
b2 = 2k2
This means that b was also an even number. Oops! This is a contradiction!
Because we supposed from the start that _ab is a fraction in simplest
_ form, and
√
now we get a and b both even. So our original statement that 2 is rational is
untrue. It must be irrational.
Activity 1
State if these numbers are rational or irrational?
_
_
√4
√3
2.
1.
4.
_
–√25
5.
_
√26
_
3.
√7
6.
√100
_
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_
1 + √2
7.
8.
3
_
√8
9.
3
_
√10
Another important number which is irrational is p.
Remember the ratio of the circumference of a circle to its diameter gives us the
quantity p.
Circumference
__ = p = 3,141592654 …
Diameter
22
or 3,14 (rounded to 2 decimals). From
Rational approximations for p are p = _
7
now on, all calculations requiring p will need you to use p on your calculator (so
hopefully you have a calculator with this button).
Now of course, irrational numbers also have to have a position on a number line
so we need to find a way to place them. Remember all your working must be
shown – you can’t just type the number into your calculator.
Activity 2
In this activity we are going to establish between which two integers any
irrational number lies. The first one has been done for you.
_
√2
1.
2.
Now 1 < 2 < 4
_
_
_
so √1 < √2 < √4
_
so 1 < √2 < 2
_
√5
3.
√69
_
●
●
●
(2 lies between the square numbers 1
and 4)
4.
_
–√24
As before, there are infinitely many irrational numbers.
_ _ 3_
A number like √2 ; √5 ; √10 is also called a SURD.
_
_
√4 or √100 is not a SURD but it is a rational number. A
By the Theorem of Pythagoras :
AC2 = 22 + 32
2
AC2 = 13
_
AC = √13
B
3
C
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_
This is the exact value for the length of AC and we should use √13 on our
calculator for any further calculations.
_
as 3,6
(correct to 1 dec. digit)
If we write √13
or as 3,61
(correct to 2 dec. digits)
or as 3,606 (correct to 3 dec. digits)
then we are using a rational approximation for a number which is actually
irrational and our final answer will not be exact.
_
Did you follow how √13 was rounded off to a specified number of decimal
digits? This is a skill you should have from junior school. You should also be able
to give answers correct to the nearest unit, ten or hundred.
Activity 3
1.
Given 23,10734569. Write this number
1.1
correct to the nearest unit
1.2
correct to the nearest ten
1.3
correct to the nearest hundred
1.4
correct to 1 dec. digit
1.5
correct to 2 dec. digits
1.6
correct to 5 dec. digits
2.
_
Round √30 to
2.1
the nearest integer
2.2
one decimal digit
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2.3
three decimal digits
We have now discussed all of the numbers in the real number system.
You should now be able to see that this system can be represented in this
picture form:
REAL NUMBER SYSTEM
– _27
Rationals
Integers
–6
400
–_
3
–7
Whole
–12
16_15
_1
–1
2
0
Natural
–5
–8
1, 2, 3, 4, 5
1,25
–2
–11
–4
0,12
–9
–10
–3
10_13
Irrationals
_
√2
_
_
√3
4
√ _12
_
_
√10
–√101
1,17326143 …
p
․
0,6
You may now be asking if any other kinds of numbers exist.
The answer is yes.
Numbers which are not real are called non-real. Since every point on a line
represents a real number, and every real number is a point of the line – this
means that non-real numbers cannot be located on a number line.
_
Any number of the form √a where a < 0 (a is a negative number) is called nonreal.
_ _ _
So √–10 ; √–16 ; √–3 are all called non-real numbers, and we cannot place them
on the real number line.
Note:
1.
2.
_
where a < 0 (a is a negative number)
_
3
√–8 = –2 is real and rational
_
3
√–9 is real but irrational
_
3
√–27 = –3 is real and rational
4 _
√a where a < 0 (a is negative number)
_ 4_ 4_
4
√–16 ; √–8 ; √–81 are all non-real
_
In general n√a is non-real only if a < 0 and n is an even number.
√a
3
You are now ready for the assessment of this section.
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Activity 4
1.
Classify (this means put into a category) the following real numbers. The
first two are done for you.
1.1
5 is a natural number, a whole number, an integer and a rational number
1.2
–10 is an integer and a rational number
1.3
_2
1.4
√144
1.5
4,37
1.6
2p
1.7
√3
_
2.
Show all working:
2.1
Express 1,24 as a common fraction
2.2
Express 0,1 + 0,02 as a common fraction
3.
__
Given p = √b2 – 4ac
3.1
Evaluate p if a = 4, b = –1 and c = –8
3.2
State whether p is rational or irrational
3
_
․
_
2
․․
․
․
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3.3
Round p off to 3 decimal digits.
3.4
Show all working to establish between which two integers p lies.
4.
_
Given E = √3x + 4
4.1
Give any value for x that would make E a rational number.
4.2
Give any value for x that would make E an irrational number.
4.3
Give any value for x that would make E a non-real number.
5.1
Choose and write down any rational number and any irrational number
between 6 and 7. Use arrows to show where they would be represented
on this number line.
5
5.2
6
7
8
Now write down a number between 6 and 7 which has a rational square
root.
6.
_
_
If x = √2 ; y = √3 and z = –16 decide whether each of the following
algebraic expressions is real and rational, real and irrational or non-real.
6.1
4x
6.2
x2
6.4
_1
6.5
√z
z
_
6.3
x + 3y
6.6
z+p
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7.
Give two different irrational numbers whose product is:.
7.1
rational
7.2
irrational
7.3
bigger than both numbers
7.4
bigger than the smaller and smaller than the bigger
7.5
smaller than both numbers
Solutions to Activities
Activity 1
1.
irrational
2.
rational
3.
irrational
4.
rational
5.
irrational
6.
rational
7.
irrational
8.
rational
9.
irrational
Activity 2
2.
Now 4 < 5 < 9
_
_
_
√4 < √5 < √9
_
2 < √5 < 3
3.
Now 64 < 69 < 81
_
_
_
√64 < √69 < √81
_
8 < √69 < 9
4.
(choose the two numbers either side of
5 which can easily be square rooted)
_
so √69 lies between 8 and 9
Be careful!
– 25 < – 24 < – 16
_
_
_
– √25 < – √24 – √16
_
– 5 < – √24 < – 4
_
so –√24 lies between –5 and –4
_ _
_
–√24
√2 √5
–5 –4 –3 –2 –1 0
1
2
3
_
√69
4
5
6
7
8
9
Activity 3
1.1
7
which is closer to 2311.
Look at 2310,7 this is 2310_
10
1.2
2310
1.3
2300
1.4
Place your pen on the digit 1 decimal place after the decimal comma.
Look to the right. This number is a 0 so we drop it and all the numbers to
the right.
So 23,1073456
1.5
®
23,1
Place your pen on the digit 2 decimals after the decimal comma. Look to
the right. This number is a 7.
Increase the digit in the second decimal place by 1. Now drop the 7 and
all the numbers to the right.
So 23,1073456
®
23,11
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1.3
Same story
So 23,1073456
2.1
2.2
2.3
®
23,10735
The rule is : if the digit to the right is a ‘5’ or larger than ‘5’(6, 7, 8, 9) then it
changes the digit to the left up by 1.
_
√30 = 5,477225575 …
_
to nearest integer √30 ≈ 5
_
√30 ≈ 5,5
_
√30 ≈ 5,477
Activity 4
2
is rational
1.3 _
3_
1.4 √144 is natural, whole, integer, rational
․
1.5
4,37 is rational
1.6
1.7
2p
is irrational
_
√
3
_ is irrational
2.1
Let x = 1,242424
2
100x = 124,242424
99x = 123
123
x=_
99
41
=_
*
2.2
․33
․
0,1 + 0,2 = 0,1111111 + 0,02222222…
= 0,1333333…
Let x = 0,13333…
10x = 1,33333…
100x = 13,3333…
* Remember the answer only will not
get you full marks/credit.
3.1
90x = 12
12 _
2
= 15
*
x=_
90__
_
p = √(–1)2 –4(4)(–8) = √129
3.2
p is irrational
3.3
11,35781669… ≈ 11,358
3.4
121 < 129 < 144
_
_
_
√121 < √129 < √144
_
11 < √129 < 12
_
∴ √129 lies between 11 and 12*
_
E = √3x + 4
_
Try x = 0 ® E = √4 = 2 this is rational
_
or try x = 4 ® E = √16 = 4 this is rational
_
or try x = 7 ® E = √25 = 5 this is rational
4.
4.1
and many more
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4.2
Could give E = 1, 2, 3, 5, 6, 7 etc
4.3
Could give E = –2, –3, –4 etc
5.
_
6
5.1
5.2
√40
6,5
7
7
Rational – choose 6 or any other (like 6_32; 6_
; 6,15)
10
_
_ _ _
Irrational – choose √40 or any other (like √41 ; √37 ; √38 )
_
25
25 _
_
<
7
and
which is rational
6<_
√ 4 = 52 _
4
169
169 _
or 6 < _
<7
and √_
= 13 which is rational
25
5
25
The only way to get these answers is by trial and error.
6.1
irrational
6.2
rational
6.3
irrational
6.4
rational
6.5
non-real
6.6
irrational
__
7.1
___
Any like √2 × √18 = 6
3
__
3
__
5
__
5
__
√2 × √4 = 2
√4 × √8 = 2
__
7.2
__
__
Any like √ 2 × √3 = √ 6
3
__
3
__
3
___
√ 5 × √ 2 = √ 10
__
7.3
__
__
Any like √2 × √3 = √ 6
_
_
__
___
_
_
√ 2 × √ 10 = √2 × √2 × √5 = 2√5
__
__
__
__
7.4
√2 _
√
√2
_
× 12 = 1 now _
< 1 < √2 etc.
2
2
7.5
√2 _
√
2
_
× 2 =_
__
__
15
or 2 × 2 = _12
3
3
__
√4
_
5
3
__
√2
_
__
__
√
√2
2 _
now _
< 2 <_
15 __5
3
__
3
3
√
√
2
4
1
now _2 < _
<_
2
2
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