,CHAPTER 1 surds-irrational numbers expressed in terms of roots
ALGEBRAIC REVIEWS of rational numbers,eg. 2;1 we;us +
. Number systems unisation
N-counting N [1,2,3 3
(2Y) x 5
Natural numbers numbers 2 2
civs 2w
= = =
-
= -
.... +
4 -
3
I
Integers -
negative numbers, natural numbers and o
conjugate/recipicol:
change sigul
x E...
= -
3; -
2; -
1;0;1;2;33 Simplification
ene
Real numbers R 2 es0
+
2
=
+
Vz(25) 2 ve.
=
+
Yes 2
=
+
5e2
↳
Rational numbers - any number that can be written
in the form where by and a bak 1.3 logarithms
a [(a,bex
=
and b = 0 it is a positive number, then the logarithm of o
can be expressed in decimal form to the base b (b >0; b =1) denoted logy, is the
↳ such that bY
terminating 0.25, 0.625 numbery
=
x
eg.
↳
non-terminating, eg. 0,5 0,3333... 5 by xforx>0
y logp
= =
= =
exponent
↳non-terminating as fractions exponent 109 base Umber
=
base =number
m e
Letx=0.53
Logarithmic rules
(b 1)
logp
logbp.log
=
Because digits repeat, multiply both sides by 100 0
=
2
100x 63.53
=
logp log by then x
y
=
=
63.53 logb*
100x x 0.63
logp(y logyx logby x
- + =
- = =
99x 63 =
logp (5) logpx-logby =
lospa b
=
..
=
=
↳ Irrational numbers
M/Q -
a real number that
can'tbe expressed as a quotient of 2 integers
eg. 4,
5
1.2 Exponents andneeds
= -
x x .x .x .x ...)
The integer is the exponent (power or index) and x is the
base...." is ith power of o r to the power of a
Exponent laws
a0 1 =
a
=Va
a
c
u
=
~m
A =CaY* (a =
=
a
a.a* a
=
c r
+
=au- it a 0
a
=
authenm=n (a) a(ab) ab
= =
,CHAPTER 2 2.2 Solving equations
_
INEARB QUADRATIC EQUATIONS Factorisation
nee
cut
factors factors
2.1 Polynomials & rational expressions e.g.2 -3x 10 =
of a of c
Enterior
>I
1x2 3x 10 0
=
- -
x +2,
. x = -2 or S
Constant function
e m e re
f(x) b
=
use
the square
↳ horizontal line e.g.x2 - 6x - 5 0
=
"i
f(z) S =
x2 -
6x 3 =
move constant to RHS
f(s) 5
=
e
- 6x E) 5 (E)
+
=
+
add () to both sides
f( z) -
5
=
(x -
3)2 =
14 solve
x
-
3 =
=
V4 square root both sides
linear
mo
e nre
function f(x) mx+C
=
x =
=
x4 3
+
isolate s
↳
Straightline :x 44 3orx 44 3
= =
+ - +
Y
1
m-gradient/slope of the line
-y x
= +
0 > 0
unequal
1
2 -
the y-intercept
7
0 0
=
unequal
rational
- O
↳theslopeor astraightlinsPass through
case, it
2.3 Simultaneous equations
↳ if L,822 lines with
are 2 non-vertical respectto Isolate a variable in the simpler equation
slopes mime, then substitute the equivalent expression for that variable
Li is
perpendicular to 12 when m,xmz=-1 into the other expression
hi parallel
is to 12 when m2
mi = Solve the
newly formed equation (note there is
only
one variable now
Que
function f(x) acch + bx
=
+ Substitute your solution(s) back into the other
↳ parabola order to find the value (5)
equation in corresponding of
the other value
P(x)
q(x) =
0
umexpressions a()
, CHAPTER 3 Graph the following as
single intervals
and
INEQUALITIES & ABSOLUTEVALUES
/
eg.(3;10]1(8;d) (4;83U[S,0)
⑳ - ⑧
3.1 Intervals 3
0 ⑧ 8
0
An interval is set of real numbers between
a
given b j
>x
lo
>
↓ ' j
numbers
↳ Interval notation :x[[8;10] ..x(4;)
3 1958
Round brackets and open circles indicate the
number is excluded while square brackets and 3.2 Absolute values
closed circles indicate the number is included the a real number
absolute value of is the distance of a
↳ Set
builder notation number on the real line (number line) from zero. So, the absolve
used to represent all sers because the set is specified value is always positive
>less than 4 less than or
equal to
Distance is positive
E
I I -
than than equal ka1 if 0
greater greater to x x
=
or
-
I
ic
↓
S12 = 2;EIR if
-
-
x co
real numbers greater than -3 and less and
represents
equal to 2 properties
m
Properties
b e
r ee n
of
inequalities ↓a) (a) =
the distance from zero to -
as a is equal
ifa s and by then as c lab) 1allbl
=
I
ifabb d
12
and cd then a c-b
+
+ =
it by o
it as and 330 then ac<bC
but ifabb and CC0 then ac <bc |a b) (a) 1b)
+
+
This is know as the triangle inequality
use
equalities
when
multiply or
dividing both sides of an Solving inequalities
em
a reere
to
less than or equal
number the direction
↳ When Ixtal Ib b3··
inequality by
=
of the the -b =x+9
a
negative
inequality sign changes eg.
12x 1) +
[
3
=
3 2x 1 =
Adding and subtracting doesn't effect the inequality 3
-
+
=
sign. -
4 2x 2
Never divide or
multiply both sides of an
inequality -
2[x[1.xd( 2;1] -
unknown
by an
↳
when lotal- 3
D
...
No solution
sand von
ALGEBRAIC REVIEWS of rational numbers,eg. 2;1 we;us +
. Number systems unisation
N-counting N [1,2,3 3
(2Y) x 5
Natural numbers numbers 2 2
civs 2w
= = =
-
= -
.... +
4 -
3
I
Integers -
negative numbers, natural numbers and o
conjugate/recipicol:
change sigul
x E...
= -
3; -
2; -
1;0;1;2;33 Simplification
ene
Real numbers R 2 es0
+
2
=
+
Vz(25) 2 ve.
=
+
Yes 2
=
+
5e2
↳
Rational numbers - any number that can be written
in the form where by and a bak 1.3 logarithms
a [(a,bex
=
and b = 0 it is a positive number, then the logarithm of o
can be expressed in decimal form to the base b (b >0; b =1) denoted logy, is the
↳ such that bY
terminating 0.25, 0.625 numbery
=
x
eg.
↳
non-terminating, eg. 0,5 0,3333... 5 by xforx>0
y logp
= =
= =
exponent
↳non-terminating as fractions exponent 109 base Umber
=
base =number
m e
Letx=0.53
Logarithmic rules
(b 1)
logp
logbp.log
=
Because digits repeat, multiply both sides by 100 0
=
2
100x 63.53
=
logp log by then x
y
=
=
63.53 logb*
100x x 0.63
logp(y logyx logby x
- + =
- = =
99x 63 =
logp (5) logpx-logby =
lospa b
=
..
=
=
↳ Irrational numbers
M/Q -
a real number that
can'tbe expressed as a quotient of 2 integers
eg. 4,
5
1.2 Exponents andneeds
= -
x x .x .x .x ...)
The integer is the exponent (power or index) and x is the
base...." is ith power of o r to the power of a
Exponent laws
a0 1 =
a
=Va
a
c
u
=
~m
A =CaY* (a =
=
a
a.a* a
=
c r
+
=au- it a 0
a
=
authenm=n (a) a(ab) ab
= =
,CHAPTER 2 2.2 Solving equations
_
INEARB QUADRATIC EQUATIONS Factorisation
nee
cut
factors factors
2.1 Polynomials & rational expressions e.g.2 -3x 10 =
of a of c
Enterior
>I
1x2 3x 10 0
=
- -
x +2,
. x = -2 or S
Constant function
e m e re
f(x) b
=
use
the square
↳ horizontal line e.g.x2 - 6x - 5 0
=
"i
f(z) S =
x2 -
6x 3 =
move constant to RHS
f(s) 5
=
e
- 6x E) 5 (E)
+
=
+
add () to both sides
f( z) -
5
=
(x -
3)2 =
14 solve
x
-
3 =
=
V4 square root both sides
linear
mo
e nre
function f(x) mx+C
=
x =
=
x4 3
+
isolate s
↳
Straightline :x 44 3orx 44 3
= =
+ - +
Y
1
m-gradient/slope of the line
-y x
= +
0 > 0
unequal
1
2 -
the y-intercept
7
0 0
=
unequal
rational
- O
↳theslopeor astraightlinsPass through
case, it
2.3 Simultaneous equations
↳ if L,822 lines with
are 2 non-vertical respectto Isolate a variable in the simpler equation
slopes mime, then substitute the equivalent expression for that variable
Li is
perpendicular to 12 when m,xmz=-1 into the other expression
hi parallel
is to 12 when m2
mi = Solve the
newly formed equation (note there is
only
one variable now
Que
function f(x) acch + bx
=
+ Substitute your solution(s) back into the other
↳ parabola order to find the value (5)
equation in corresponding of
the other value
P(x)
q(x) =
0
umexpressions a()
, CHAPTER 3 Graph the following as
single intervals
and
INEQUALITIES & ABSOLUTEVALUES
/
eg.(3;10]1(8;d) (4;83U[S,0)
⑳ - ⑧
3.1 Intervals 3
0 ⑧ 8
0
An interval is set of real numbers between
a
given b j
>x
lo
>
↓ ' j
numbers
↳ Interval notation :x[[8;10] ..x(4;)
3 1958
Round brackets and open circles indicate the
number is excluded while square brackets and 3.2 Absolute values
closed circles indicate the number is included the a real number
absolute value of is the distance of a
↳ Set
builder notation number on the real line (number line) from zero. So, the absolve
used to represent all sers because the set is specified value is always positive
>less than 4 less than or
equal to
Distance is positive
E
I I -
than than equal ka1 if 0
greater greater to x x
=
or
-
I
ic
↓
S12 = 2;EIR if
-
-
x co
real numbers greater than -3 and less and
represents
equal to 2 properties
m
Properties
b e
r ee n
of
inequalities ↓a) (a) =
the distance from zero to -
as a is equal
ifa s and by then as c lab) 1allbl
=
I
ifabb d
12
and cd then a c-b
+
+ =
it by o
it as and 330 then ac<bC
but ifabb and CC0 then ac <bc |a b) (a) 1b)
+
+
This is know as the triangle inequality
use
equalities
when
multiply or
dividing both sides of an Solving inequalities
em
a reere
to
less than or equal
number the direction
↳ When Ixtal Ib b3··
inequality by
=
of the the -b =x+9
a
negative
inequality sign changes eg.
12x 1) +
[
3
=
3 2x 1 =
Adding and subtracting doesn't effect the inequality 3
-
+
=
sign. -
4 2x 2
Never divide or
multiply both sides of an
inequality -
2[x[1.xd( 2;1] -
unknown
by an
↳
when lotal- 3
D
...
No solution
sand von
