Algebra
Algebraic Notation
Algebraic notation is a way of expressing mathematical ideas concisely. In
algebra, letters are often used in equations as unknown numbers.
Usually, in an equation you could see 6y to mean 6 x y or ‘6 multiplied by y’.
We leave out the multiplication sign to make the equations look more
elegant and neater when dealing with them.
Example
It is much easier to deal with an equation that looks like (1) as opposed to
(2), even though they both mean the same thing:
(1) 6a+3b+7c=0
(2) 6xa+ 3xb+7xc=0
Substitution
Substitution is putting numbers in place of letters to calculate the value of
an expression.
Example
Let e=3 and f=4.
What is that value of 40ef?
From above, we know that 40ef is the same as writing 40 multiplied by e
multiplied by f.
, First of all, as we aware that e=3, we replace e with 3 in the expression and
we are left with: 40 multiplied by 3 multiplied by f.
Likewise, we do the same with f. We replace f with 4 in the expression and
we are left with: 40 multiplied by 3 multiplied by 4.
Therefore, 40ef= 40x3x4=280.
Algebraic Vocabulary
In order to fully grasp the concepts that we deal with in the topic of
Algebra, we need to understand a handful of key terms:
Constant- A value or number that never changes in an equation e.g. 4
is a constant because it is a value that doesn’t change.
Equation- A grouping of terms or values that use a sign to show an
equal relationship e.g. 69ab=69 is an equation.
Exponent- A small number written on the right-hand side, above a
number or variable often referred to as a ‘power’. It could also be
written as a^b where b is the exponent e.g. in the term 5^2, 2 is the
exponent.
Expression- Any combination of values and operations that can be
used to show how things belong together and compare to one
another e.g. 80ef+4g is an expression. Always remember, expressions
do not have an equal sign!
Factor- To change two or more terms to just one term so that you
can perform other processes e.g. 20a+6b+50c=2(10a+3b+25c).
Operation- An action performed upon one or two numbers to
produce a resulting number e.g. multiplication, addition, subtraction,
division, square root, and more.
Simplify- To combine all that can be combined by collecting like-
terms, putting the equation its simplest form e.g.
11a+3a+6a+3b=20a+3b.
Solve- To work out or find the answer to.
Term- A grouping together of one or more factors which include
numbers and/or variables e.g. 5tu is a term and the expression 5tu+9
has 2 terms.
Variable- Often referred to as letters that are represented as
unknown numbers e.g. in the term 24t, t is the variable as it is a value
that does not have a fixed value.
,Algebraic Expressions
An algebraic expression is an expression made up of a combination of
numbers, variable separating each term
Mathematical Formulae
A mathematical formula is a mathematical rule that is followed to work out
a specific concept. These formulae are filled with unknown variables with
each one standing for something different.
In Maths, we have a few key formulae that are used pretty frequently e.g.
the quadratic formula, formula for the area of a triangle, the sine and
cosine rule and more.
Quadratic formula of ax^2+bx+c=0:
Area of a triangle:
Sine and Cosine rule:
Example
Solve the following quadratic equation using the quadratic formula:
x^2+5x+6=0
, In this example a=1, b=5 and c=6.
Plugging these into the equation, we end up with:
x=-5+(sqrt(5^2-4(1)(6)))/2(1)
=
Algebraic Equivalence and Proof
Algebraic equivalence are rules applied in Maths to show that when two
things are added or multiplied together, usually the relationship between
even and odd numbers, a specific outcome is given.
We prove all of these in a similar way.
To represent an even number, it is indicated as 2x, where x is any natural
number. To represent an odd number, it is indicated as 2x+1, where x is any
natural number.
To prove even+even=even, we take two even numbers, namely 2a and 2b,
where a and b are natural numbers.
2a+2b=2(a+b).
As we can factor a two out of the sum of 2a and 2b, it shows that it is also
even and therefore it has been proven.
Key fact: proofs are not proofs if you try and use an example to prove a
concept!
Algebraic Notation
Algebraic notation is a way of expressing mathematical ideas concisely. In
algebra, letters are often used in equations as unknown numbers.
Usually, in an equation you could see 6y to mean 6 x y or ‘6 multiplied by y’.
We leave out the multiplication sign to make the equations look more
elegant and neater when dealing with them.
Example
It is much easier to deal with an equation that looks like (1) as opposed to
(2), even though they both mean the same thing:
(1) 6a+3b+7c=0
(2) 6xa+ 3xb+7xc=0
Substitution
Substitution is putting numbers in place of letters to calculate the value of
an expression.
Example
Let e=3 and f=4.
What is that value of 40ef?
From above, we know that 40ef is the same as writing 40 multiplied by e
multiplied by f.
, First of all, as we aware that e=3, we replace e with 3 in the expression and
we are left with: 40 multiplied by 3 multiplied by f.
Likewise, we do the same with f. We replace f with 4 in the expression and
we are left with: 40 multiplied by 3 multiplied by 4.
Therefore, 40ef= 40x3x4=280.
Algebraic Vocabulary
In order to fully grasp the concepts that we deal with in the topic of
Algebra, we need to understand a handful of key terms:
Constant- A value or number that never changes in an equation e.g. 4
is a constant because it is a value that doesn’t change.
Equation- A grouping of terms or values that use a sign to show an
equal relationship e.g. 69ab=69 is an equation.
Exponent- A small number written on the right-hand side, above a
number or variable often referred to as a ‘power’. It could also be
written as a^b where b is the exponent e.g. in the term 5^2, 2 is the
exponent.
Expression- Any combination of values and operations that can be
used to show how things belong together and compare to one
another e.g. 80ef+4g is an expression. Always remember, expressions
do not have an equal sign!
Factor- To change two or more terms to just one term so that you
can perform other processes e.g. 20a+6b+50c=2(10a+3b+25c).
Operation- An action performed upon one or two numbers to
produce a resulting number e.g. multiplication, addition, subtraction,
division, square root, and more.
Simplify- To combine all that can be combined by collecting like-
terms, putting the equation its simplest form e.g.
11a+3a+6a+3b=20a+3b.
Solve- To work out or find the answer to.
Term- A grouping together of one or more factors which include
numbers and/or variables e.g. 5tu is a term and the expression 5tu+9
has 2 terms.
Variable- Often referred to as letters that are represented as
unknown numbers e.g. in the term 24t, t is the variable as it is a value
that does not have a fixed value.
,Algebraic Expressions
An algebraic expression is an expression made up of a combination of
numbers, variable separating each term
Mathematical Formulae
A mathematical formula is a mathematical rule that is followed to work out
a specific concept. These formulae are filled with unknown variables with
each one standing for something different.
In Maths, we have a few key formulae that are used pretty frequently e.g.
the quadratic formula, formula for the area of a triangle, the sine and
cosine rule and more.
Quadratic formula of ax^2+bx+c=0:
Area of a triangle:
Sine and Cosine rule:
Example
Solve the following quadratic equation using the quadratic formula:
x^2+5x+6=0
, In this example a=1, b=5 and c=6.
Plugging these into the equation, we end up with:
x=-5+(sqrt(5^2-4(1)(6)))/2(1)
=
Algebraic Equivalence and Proof
Algebraic equivalence are rules applied in Maths to show that when two
things are added or multiplied together, usually the relationship between
even and odd numbers, a specific outcome is given.
We prove all of these in a similar way.
To represent an even number, it is indicated as 2x, where x is any natural
number. To represent an odd number, it is indicated as 2x+1, where x is any
natural number.
To prove even+even=even, we take two even numbers, namely 2a and 2b,
where a and b are natural numbers.
2a+2b=2(a+b).
As we can factor a two out of the sum of 2a and 2b, it shows that it is also
even and therefore it has been proven.
Key fact: proofs are not proofs if you try and use an example to prove a
concept!
