Download Rational and Irrational Numbers Notes

Chapter 1: Rational & Irrational Numbers
Exercise 1A – Operations with Fractions
Real Numbers
We use numbers such as integers, fractions and decimals everyday. They form part of what
is call the Real Number System. Real numbers can be divided into two categories – rational
numbers and irrational numbers
Rational numbers – integers, fractions, finite decimals, recurring decimals
Irrational numbers – non recurring and non termination decimals, surds, special
numbers π and е
Simplifying Fractions
Fractional answers should be expressed in simplest form. This is done by dividing the
numerator and the denominator by their highest common factor (HCF).
Example 1: Write 25/60 in simplest form
a) Write fraction and divide by HCF
25/60
HCF = 5
(25/5) / (60/5) = 5/12
Using the Four Operations with Fractions
Addition and Subtraction
1. When adding and subtracting write each fraction with the same denominator.
Common denominator is the lowest common multiple (LCM).
2. When adding or subtracting mixed number, change to an improper fraction then
continue with step 1.
Example 2: Evaluate 3/5 + 3/6
a) Write expression
3/5 + 5/6
b) Write each fraction with the same denominator
3/5 + 3/6
6 x (3/5) + 5 x (3/6)
18/30 + 15/30
c) Add fractions by adding numerators
33/30
d) Simplify by writing as a missed number
1 3/30 = 1 1/10
Example 3 – Evaluate 3 ½ - 1 3/5
a) Write expression
3 ½ - 1 3/5
b) Change each to an improper fraction
3 ½ = 3 x 2 + 1 = 7/2
1 3/5 = 1 x 5 + 3 = 8/5
7/2 – 8/5
c) Write both fractions with the same denominator
7/2 – 8/5
5 x (7/2) – 2 x (8/5)
35/10 – 16/10
d) Subtract 2nd fraction from 1st
35/10 – 16/10
19/10
e) Simplify
1 9/10
Multiplication and Division
1. When multiplying fractions, cancel if appropriate, multiply the numerators and then
multiply the denominators.
2. When dividing fractions, change the division sign to a multiplication sign, and reverse
the second fraction, then follow the rules of multiplying.
3. Change missed numbers to improper fractions before x or ÷.
Example 4: Evaluate 2/5 x 5/6
a) Write expression
2/5 x 5/6
b) Cancel or divide numerators and denominators by same number where applicable
2/5 x 5/6
2/2
5/5
5/5
6/2
= 1/1 x 1/3
c) Multiply together
1/1 x 1/3
(1x1)/(1x3)
1/3
Example 5: Evaluate 2 1/3 ÷ ¼
a) Write expression
2 1/3 ÷ ¼
b) Change mixed numbers to improper fractions
2 1/3 = 2 x 3 +1 = 7/3
c) Change ÷ to x and swap 2nd fraction
7/3 ÷ ¼
7/3 x 4/1
d) Multiply together
7/3 x 4/1
(7x4)/(3x1)
28/3
e) Changed to mixed numbers
28/3 = 9 1/3
Exercise 1B – Finite and Recurring Decimals
Recurring decimals: repeat the last decimal places over and over again.
: represented by a bar or dots placed over the recurring numbers.
Example 1: Express the following as a recurring decimal: 7/12
a) Write fraction
7/12
b) Divide numerator by denominator until recurring pattern occurs.
0.58333
12 7.0100404040
c) Write the answer
7/12 = 0.583
To convert recurring decimals to fractions requires some algebraic skills.
Example 2: Convert 0.63 to fraction
a) Write recurring decimal and expanded form
0.63 = 0.636363...
b) Let x equal expanded form and call it equation [1]
x = 0.636363...
[1]
c) Multiply both sides of equation [1] by 100 because there are two repeating numbers,
call this equation [2]
x = 0636363...
[1]
100 x x = 100 x 0.63663
100x = 63.636363...
[2]
d) Subtract [1] from [2] to remove recurring part of decimal
[2] – [1] = 100x –x = 63.636363 – 0.636363
99x = 63
e) Solve equation and write in simplest from
99x = 63
X = 63 / 99
X = 7/11
Class Worked Example
Example 3: Convert 0.93 to fraction.
a) 0.93 = 0.93333....
b) x = 0.93333... [1]
c) x10
x = 0.9333...
10 x x = 10 x 0.9333...
10x = 9.3333 [2]
d) [2] – [1]
10x –x = 9.33333 – 0.9333
9x = 8.4
e) 9x = 0.84
f) x10 to get rid of decimal
x = 84/90
x = 28/30
Exercise 1C – Irrational Numbers
Irrational numbers cannot be expressed as fractions. These include:
1. Non recurring, infinite (non terminating) decimals
2. The special numbers π and е
3. Surds or roots of numbers that do not have a finite, exact answer, for example √5
and 3√6
A surd is an exact answer but the calculator answer is an approximation as it has been
rounded.
Example 1: State whether the following a surd or not.
(1) √3
a) Write the number and consider square roots which can be evaluated, √1 = 1,
√4 =2
b) Check on calculator to see if an exact number is the answer, if necessary
√3 is a surd.
(2) 3√8
a) Write number, consider whether the cube root can be found by cubing small
numbers and write the exact answer if there is one.
3√8
1x1x1=1
2x2x2=8
3√8
= 2, therefore it is not a surd.
Rational Approximations for Surds
When an infinite number is rounded, the answer is not exact but it is very close to the actual
value of the number. It is called a rational approximation because once it is rounded it
becomes finite and therefore rational. Use a calculator and round to 2 decimal places for
these questions.
Exercise 1D – Simplifying Surds
Some surds can be reduced to simplest form. Only square roots will be considered in this
section.
Consider √36 = 6
Now 36 = 9x4, so we could say;
√(9x4) = 6
Taking √9 and √4 separately
√9 x √4 = 3 x 2 = 6
If both √(9x4) ad √9 x √4 = 6 then, √9x4 = √9 x √4
This can be stated as √ab = √a x √b and can be used to simplify surds.
A surd can be simplified by dividing it into 2 square roots, one of which is the highest perfect
square that will divide evenly into the original number.
Example 1: Simplify √18
a) Write the surd and divide it into 2 parts, one being the highest perfect square that
will divide into the surd
√18 = √(9x2)
= √9 x √2
b) Write in simplest form by taking the square root of the perfect square.
√9 x √2 = 3 x √2 = 3√2
Sometimes it is necessary to change a simplified surd to a whole surd. The reverse process is
applied.
Example 2: Write 4√5 in the form √a
a) Write the expression as a produce of an integer and a surd
4√5 = 4 x √5
b) Square the whole number part then express the whole number as a square root
4 x √5
= 42 x √5
= √42 x √5
c) Write the simplest surd and express it as the product of 2 square roots
√16 x √5
d) Multiply the square roots to give a single surd
√16 x √5
= √80
Exercise 1E – Addition & Subtraction of Surds
Operations with surds have the same rules as operations with algebra.
1. Like surds are those which containing the same surd when written in simplest form.
2. Like surds can be added or subtracted after they have been written in simplest form.
Example 1: Simplify 6√3 + 2√3 + 4√5 - 5√5
a) Write the expression
6√3 + 2√3 + 4√5 - 5√5
b) All surds are in simplest form, so collect like surds
8√3 -√5
We need to check that all surds are fully simplified before we can be sure whether or not
they can be added or subtracted as like terms.
Class Worked Example
Example 2: Simplify 5√75 - 6√12 + 2√8 + 4√3
a) Write the expression
5√75 - 6√12 + 2√8 + 4√3
b) Simplify all surds
(5 x √25 x √3) – (6 x √4 x √3) + (2 x √4 x √2) + (4 x √3)
= (5 x 5 x √3) = (6 x 2 x √3) + (2 x 2 x √2) + (4 x √3)
= (25 x √3) – (12 x √3) + (4 x √2) + (4 x √3)
= 25√3 - 12√3 + 4√2 + 4√3
c) Collect like surds
= 17√3 + 4√2
Exercise 1F – Multiplication and Division of Surds
Surds can be multiplied and divided the same way as pronumerals in algebra.
1) To multiply and divide surds, use the following
a) √a x √b = √ab
b) c√a x d√b = cd√ab
c) (√a/√b) = √a/b
2) Leave answers in simplest surd form
3) To remove a bracket containing surds, multiply each term outside the bracket by
each term inside the bracket
4) To expand two brackets containing surds, multiply each term in the first bracket by
each term in the second bracket
Example 1: Evaluate -4√5 x 7√6
a) Write the expression, multiply whole numbers and multiply the surds
-4√5 x 7√6 = -4 x 7 x √5 x √6
b) Simplify if appropriate
-28√30
Example 2: Evaluate (16√15) / (24√75)
a) Write the expression and simplify denominator
(16√15) / (24√75) = (16√15) / (24√25x3) = (16√15) / (24x5√3)
16/24 = 2/3
2√15 / (3x5√3)
b) Simplify the fraction formed by the rational and irrational parts separately.
2√15 / (3 x 5√3)
=2/15 x √(15/3)
= (2√5)/15
Class Worked Example
Example 3: Expand (√2 + √6)(2√3 - √6)
a) Write expressions
(√2 + √6)(2√3 - √6)
b) Multiply each term in the first bracket by each term in the second bracket (F.O.I.L)
(√2 + √6)(2√3 - √6)
= (√2 x 2√3) + (√2 x -√6) + (√6 x 2√3) + (√6 x -√6)
= 2√6 - √12 + 2√18 - √36
c) Simplify surds
= 2√6 - √12 + 2√18 - √36
= 2√6 - √(4x3) + 2√(9x2) – 6
= 2√6 - 2√3 + 2x3√2 – 6
= 2√6 - 2√3 + 6√2 - 6
Exercise 1G – Writing Surds Fractions with a Rational Denominator
1/√2 is a fraction with a surd in the denominator. If we multiply 1/√2 by 1 its values will
remain unchanged. If the numerator and the denominator are both multiplied by the same
number, the value of the fraction stays the same because we are multiplying by 1.
Example 1: Express the following in simplest form with a rational denominator – 1/√5
a) Write the fraction
1/√5
b) Multiply the numerator and denominator by the surd in the denominator
(1/√5) x (√5/√5)
= √5 /√25
c) Simplify
√5 / √25 = √5/5
If there is a binomial denominator (two terms) such as (3 + √2) then the fraction can be
written with a rational denominator by multiplying the numerator and denominator by the
same expression with the opposite sign.
Example 2: Express 5/(2+√3) in simplest form with a rational denominator
a) Write fraction
5 / (2+√3)
b) Multiply both the numerator and denominator by (2-√3)
= (5/(2+√3) x ((2-√3)/(2-√3))
= (5x(2-√3))/ (2+√3)(2-√3)
= (5(2√3))/((2x2)+(2x-√3)+( √3x2)+( √3-√3))
= 5(2-√3) / (4 - 2√3 + 2√3 - √9)
= 5(2-√3) / (4-3)
= 5(2-√3) / 1
= 5(2-√3)