Rational and Irrational Numbers

1
Rational and Irrational Numbers
10
Mathematics in Workplaces
10.1 Square Roots
10.2 Rational and Irrational Numbers
10.3 Surds
10.4 Operations of Surds
Chapter Summary
2
Mathematics in Workplaces
Artist Some artists believe that objects with
shapes containing the golden ratio (1 1.618...)
are particularly beautiful to the human eye.
Artists love to apply the ratio to their works
such as the painting of the Mona Lisa by Leonardo
Da Vinci.
For example, the ratio of the length and width of
each of the rectangles (as shown in the above
painting) is 1 1.618.
3
10.1 Square Roots
Consider the square numbers 1, 4, 9, 16, 25, 36,
.
We know that
1 ? 1 ? 12 ? 1 2 ? 2 ? 22 ? 4 3 ? 3 ? 32 ?
9 ? 9 ? 9 ? 92 ? 81 ?
The number 81 is called the square of 9.
9 is called the square root of 81 and we express
it as .
__ The symbol ? is
called the radical sign.
4
10.1 Square Roots
If x2 ? y, then x is a square root of y.
Note that (?4)2 ? 16 and (?4)2 ? 16.
Therefore, 16 has 2 square roots, ?4 and ?4.
5
10.1 Square Roots
Example 10.1T
Find the square roots of each of the following
numbers. (a) 64 (b) 0.81 (c)
Solution
(a) ? 8 ? 8 ? 64 and (?8) ? (?8) ? 64
? The square roots of 64 are ?8.
(b) ? 0.9 ? 0.9 ? 0.81 and (?0.9) ?
(?0.9) ? 0.81
? The square roots of 0.81 are ?0.9.
(c) ?
3 ? 3 ? 9 and (?3) ? (?3) ? 9
? The square roots of are ?3.
6
10.1 Square Roots
For the square root of a number that cannot be
expressed as an integer, a fraction or a decimal,
we can find its value using a calculator.
For example, to find the positive square root of
2.85, key in
2.85
v
EXE
We obtain the positive square root of 2.85 is
1.69. (cor. to 3 sig fig.)
A calculator can only find the positive square
root of a number.
7
10.2 Rational and Irrational Numbers
A. Rational Numbers
Numbers that can be expressed in the form ,
where x, y are integers and y ? 0 are called
rational numbers.
These include integers, fractions, terminating
decimals and recurring decimals.
1. Integers For example, 2 and 8 can be
expressed as and respectively.
2. Proper fractions, improper fractions and mixed
numbers For example, can be expressed as
.
8
10.2 Rational and Irrational Numbers
A. Rational Numbers
3. Terminating decimals Decimals with a limited
number of non-zero digits after the decimal point.
For example, 1.2 and 4.5 can be expressed as
and respectively.
4. Recurring decimals Decimals with a pattern of
digits that repeat indefinitely after the decimal
point.
.
For example, convert 0.3 into a fraction
.
Let x ? 0.33333... .......... (1)
Then 10x ? 3.33333... .......... (2)
(2) ? (1) 10x ? x ? 3.33333... ? 0.33333...
9x ? 3
9
10.2 Rational and Irrational Numbers
A. Rational Numbers
Example 10.2T
Express the following numbers in the form ,
where x and y are integers and y ?
0. (a) 7 (b) (c) 3.55
Solution
10
10.2 Rational and Irrational Numbers
A. Rational Numbers
Example 10.3T
. .
Express 0.45 in the form , where x and y are
integers and y ? 0.
Solution
Let x ? 0.45454545... ............ (1)
Then 100x ? 45.45454545... .......... (2)
(2) ? (1) 100x ? x ? 45.45454545... ?
0.45454545...
99x ? 45
11
10.2 Rational and Irrational Numbers
B. Irrational Numbers
Numbers that cannot be expressed in the form
, where x, y are integers and y ? 0 are called
irrational numbers.
For example, (? 2.44948...) and (?
3.87298...) are irrational numbers, they are
neither terminating decimals nor recurring
decimals.
The golden ratio mentioned earlier in this
chapter is also an irrational number.
Golden ratio ? 1 1.618... ?
12
10.2 Rational and Irrational Numbers
B. Irrational Numbers
Example 10.4T
Solution
13
10.3 Surds
Square roots that are irrational numbers are
called surds.
1. Since and are irrational numbers,
they are surds.
2. Since (? 2) and are
rational numbers after simplification, they are
not surds.
3. p is an irrational number, but it does not
contain a radical sign. Therefore, it is not a
surd.
14
10.3 Surds
A. Values of Surds
We can estimate the value of the positive square
root of a non-square number without using a
calculator.
For example, to estimate
? is greater than 3 and less than 4.
We can also locate the position of an irrational
number on a number line. This will be discussed
further in the next chapter.
15
10.3 Surds
B. Properties of Surds
When simplifying a surd, we express the number as
the product of its prime factors in index
notation.
For example,
16
10.3 Surds
B. Properties of Surds
Example 10.5T
Solution
(a) 12 ? 2 ? 2 ? 3
75 ? 3 ? 5 ? 5
? 22 ? 3
? 3 ? 52
(b)
17
10.3 Surds
B. Properties of Surds
Example 10.6T
Express in its simplest form.
Solution
18
10.4 Operations of Surds
A. Addition, Subtraction, Multiplication and
Division of Surds
When expressing surds in their simplest forms,
those with the same number inside the radical
signs are called like surds.
Those with different numbers inside the radical
signs are called unlike surds.
For example
When doing calculations with surds, only like
surds can be added or subtracted.
19
10.4 Operations of Surds
A. Addition, Subtraction, Multiplication and
Division of Surds
Example 10.7T
Solution
20
10.4 Operations of Surds
A. Addition, Subtraction, Multiplication and
Division of Surds
Example 10.8T
Solution
21
10.4 Operations of Surds
A. Addition, Subtraction, Multiplication and
Division of Surds
Example 10.9T
Solution
22
10.4 Operations of Surds
A. Addition, Subtraction, Multiplication and
Division of Surds
Example 10.10T
Solution
23
10.4 Operations of Surds
B. Rationalization of Denominators
Consider the surd .
The denominator is an irrational number.
The process of converting a denominator from an
irrational number to a rational number without
changing its value is called rationalization.
24
10.4 Operations of Surds
B. Rationalization of Denominators
Example 10.11T
Solution
25
10.4 Operations of Surds
B. Rationalization of Denominators
Example 10.12T
Simplify .
Solution
26
Chapter Summary
10.1 Square Roots
1. If x2 ? y, then x is a square root of y.
27
10.2 Rational and Irrational Numbers
Chapter Summary
Numbers that can be expressed in the form ,
where x, y are integers and y ? 0 are called
rational numbers.
Other numbers are called irrational numbers.
28
10.3 Surds
Chapter Summary
Irrational numbers that contain radical signs are
called surds.
29
10.4 Operations of Surds
Chapter Summary
Surds with the same number inside the radical
signs are called like surds.
Otherwise, they are called unlike surds.
When doing calculations with surds, only like
surds can be added or subtracted.
The process of converting a denominator from an
irrational number to a rational number without
changing its value is called rationalization.
30
10.1 Square Roots
Follow-up 10.1
Find the square roots of each of the following
numbers. (a) 36 (b) 2.25 (c)
Solution
(a) ? 6 ? 6 ? 36 and (?6) ? (?6) ? 36
? The square roots of 36 are ?6.
(b) ? 1.5 ? 1.5 ? 2.25 and (?1.5) ?
(?1.5) ? 2.25
? The square roots of 2.25 are ?1.5.
31
10.2 Rational and Irrational Numbers
A. Rational Numbers
Follow-up 10.2
Express the following numbers in the form ,
where x and y are integers and y ?
0. (a) ?3 (b) (c) 0.96
Solution
32
10.2 Rational and Irrational Numbers
A. Rational Numbers
Follow-up 10.3
. .
Express 1.27 in the form , where x and y are
integers and y ? 0.
Solution
Let x ? 1.27272727... .............. (1)
Then 100x ? 127.27272727... .......... (2)
(2) ? (1) 100x ? x ? 127.27272727... ?
1.27272727...
99x ? 126
33
10.2 Rational and Irrational Numbers
B. Irrational Numbers
Follow-up 10.4
Solution
(c) ? p ? 1 ? 4.141592... , which is neither
terminating nor recurring.
? p ? 1 is an irrational number.
34
10.3 Surds
B. Properties of Surds
Follow-up 10.5
Solution
(a) 27 ? 3 ? 3 ? 3
48 ? 2 ? 2 ? 2 ? 2 ? 3
? 33
? 24 ? 3
(b)
35
10.3 Surds
B. Properties of Surds
Follow-up 10.6
Express in its simplest form.
Solution
36
10.4 Operations of Surds
A. Addition, Subtraction, Multiplication and
Division of Surds
Follow-up 10.7
Solution
37
10.4 Operations of Surds
A. Addition, Subtraction, Multiplication and
Division of Surds
Follow-up 10.8
Solution
38
10.4 Operations of Surds
A. Addition, Subtraction, Multiplication and
Division of Surds
Follow-up 10.9
Solution
39
10.4 Operations of Surds
A. Addition, Subtraction, Multiplication and
Division of Surds
Follow-up 10.10
Solution
40
10.4 Operations of Surds
B. Rationalization of Denominators
Follow-up 10.11
Solution
41
10.4 Operations of Surds
B. Rationalization of Denominators
Follow-up 10.12
Simplify .
Solution