Surds And Indices Page 22
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28Some properties of real numbers
I
Exercise 11I
1
Use the fraction key on your calculator to evaluate (as fractions) the following.
−1
−1
−1
−1
−1
5 __
6 __
9 __
2 __
3 __
(
)
(
)
(
)
(
)
(
)
a
b
c
d
e
2
5
7
3
4
a __
−1
b __
(
)
=
2
Show that
a .
b
3
If a and b are real numbers, determine whether the following
A counter example is an example that
demonstrates the statement is false.
statements are true or false. If false, give a counter example.
(Hint: Try several diff erent pairs of real numbers to test the truth of the statements.)
a + b is a real number.
a − b is a real number.
a
i
ii
a × b is a real number.
a ÷ b is a real number.
iii
iv
a + b = b + a
a − b = b − a
b i
ii
a × b = b × a
a ÷ b = b ÷ a
iii
iv
(a + b) + c = a + (b + c)
(a − b) − c = a − (b − c)
c
i
ii
(a × b) × c = a × (b × c)
(a ÷ b) ÷ c = a ÷ (b ÷ c)
iii
iv
a × 0 = 0
a + 0 = a
d i
ii
a × 1 = a
a ÷ a = 1
e
i
ii
4
If m and n are rational numbers, determine whether the following statements are true or false. If false give
a counter example.
m + n is always rational.
m − n is always rational.
a
b
m × n is always rational.
m ÷ n is always rational.
c
d
5
Find a pair of surds that satisfy each condition.
Remember:
× gives product.
a
The product of the surds is irrational.
÷ gives quotient.
b
The product of the surds is rational.
c
The quotient of the surds is irrational.
d
The quotient of the surds is rational.
6 a
Write three consecutive integers starting with y.
Consecutive means in order
and without gaps.
b
Hence show that the sum of any three consecutive
integers is divisible by 3.
7 a
Show that any even real number can be written in the form 2k, where k is an integer.
Show that any odd real number can be written in the form 2k + 1, where k is an integer.
b
c
Hence prove the following properties of real numbers.
i
The sum of any two even numbers is even.
ii
The sum of any two odd numbers is even.
iii
The sum of an even number and an odd number is odd.
iv
The product of two even numbers is even.
v
The product of an odd number and an even number is even.
vi
The product of two odd numbers is odd.
22
Insight Maths 9
Australian Curriculum
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