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Title: Algebraic order of operation notes
Description: Algebraic number theory is a branch of mathematics that studies the properties of algebraic numbers, which are complex numbers that are roots of polynomials with integer coefficients. Algebraic numbers include rational numbers, irrational numbers such as square roots, cube roots, and other radicals, and complex numbers that are algebraic extensions of the rationals. Algebraic number theory studies the algebraic properties of these numbers, such as their algebraic degree, which is the degree of the polynomial that they satisfy, and their algebraic conjugates, which are the other roots of the polynomial.
Description: Algebraic number theory is a branch of mathematics that studies the properties of algebraic numbers, which are complex numbers that are roots of polynomials with integer coefficients. Algebraic numbers include rational numbers, irrational numbers such as square roots, cube roots, and other radicals, and complex numbers that are algebraic extensions of the rationals. Algebraic number theory studies the algebraic properties of these numbers, such as their algebraic degree, which is the degree of the polynomial that they satisfy, and their algebraic conjugates, which are the other roots of the polynomial.
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Faculty of Mathematics
Waterloo, Ontario N2L 3G1
Grade 7 & 8 Math Circles
October 8/9, 2013
Algebra
Introduction
When evaluating mathematical expressions it is important to remember that there is only
one right answer
...
Order of Operations
When working on more complicated mathematical expressions that involve multiple operations and many steps it is very important to do the math in the right order to ensure that
you get the right answer
...
Begin by simplifying everything inside of brackets
...
Evaluate anything with exponents
...
Starting from the left, work out all of the multiplication and division, whichever
comes first
...
Starting from the left, finish by evaluating all of the addition and subtraction,
whichever comes first
...
Similarly, addition is no more important than subtraction
...
(a) 15 ÷ 3 − 1
(b) 7 × 3 + 4
(c) 48 ÷ 4 + 4 × 5
(d) 48 ÷ (4 + 4) × 5
(e) 11 − 62 ÷ 12 + (4 + 5 × 2) ÷ 7
Distributive Property
The Distributive Property is used when multiplying a sum by a single term or another sum
...
That is,
a × (b + c) = a × b + a × c
...
To remember this, the mnemonic FOIL is used
...
Outsides Multiply the two terms on the outsides (a and d)
...
Lasts
Multiply the last term in each sum (b and d)
...
• a positive multiplied or divided by a positive will result in a positive number
• a negative multiplied or divided by a negative will result in a positive number
• a positive multiplied or divided by a negative will result in a negative number
Example
(2 + 5) × (6 − 3) = (2)(6) + (2)(−3) + (5)(6) + (5)(−3)
= 12 − 6 + 30 − 15
= 21
2
5
6
12
30
−3
−6
−15
12 + 30 − 6 − 15 = 21
We can check that this is right by using our BEDMAS rules:
(2 + 5) × (6 − 3) = 7 × 3
= 21
3
Exercises II
Evaluate the following expressions without a calculator
...
That is, factoring breaks down an expression into
the product of its simplest components
...
The number 21 can be written as 21 = 3 × 7
...
Because 3 and 7 are both prime numbers we call this a prime factorization
...
The number 48 can be written as 48 = 4 × 12
...
12 = 2 × 6
=2×2×3
4=2×2
Notice that 2 and 3 are both prime numbers
...
You can factor a common divisor out of a sum: ab + ac = a(b + c)
...
6 and 10 are both even numbers and therefore divisible by 2
...
(a) 35
(b) 36
(c) 144
Factor a common divisor out of the following sums
...
How old is he right now?
How can you solve this problem?
You could “guess and check”, where you start guessing numbers and check to see if they
satisfy the conditions
...
Instead, you can set up an equation with a variable representing Matt’s age right now
...
In algebra it is often our goal to
isolate a variable so that it is no longer unknown
...
Determine what you are trying to isolate/solve for
...
Simplify the equation as much as possible by adding and subtracting like terms
...
You can think of it like adding apples and oranges
...
Another, more
mathematical, example:
5 + x + 3y − 2 − y + 2x = 3 + 3x + 2y
3
...
The goal of
isolating a variable, say x, is to obtain the form x =
...
= x
...
There should be no other xs on the other side of the
equal sign
...
When performing opposite operations, what you do to one side of the equation
you must do to the other side of the equation
...
1
...
3
...
2x = 10
10
2x
=
2
2
x=5
5
...
6
x + 3 = 10
x + 3−3 = 10−3
x=7
4x − 1 = 7
4x − 1+1 = 7+1
4x
8
=
4
4
x=2
7
...
x + 10 = 2(x − 6)
x + 10+12 = 2x − 12+12
x + 22−x = 2x−x
22 = x
Matt is 22
...
Show all of your steps
...
(1) She can buy one deck of Pok´emon cards and one Rubik’s Cube for $20
...
What are the prices of a deck of Pok´emon cards and a Rubik’s Cube?
How can you solve this problem?
Notice that we have two unknowns: the price of a deck of Pok´emon cards (x) and the price
of a Rubik’s Cube (y)
...
Mathematically, 1x + 1y = 20
...
Mathematically, 2x = 7 + 1y or 2x − 1y = 7
...
A System of Equations is a set of multiple equations dealing with multiple variables
...
7
Example
A system of equations looks like:
x + y = 20
2x − y = 7
(1)
(2)
We will use everything we have covered today, plus substitution, to solve this system
...
Steps for Solving:
1
...
2
...
3
...
This will become more clear in the following example
...
x + y−x = 20−x
y = 20 − x
Now replace y in (2) with 20 − x
...
2x − (20 − x) = 7
2x − 20 + x = 7
3x − 20+20 = 7+20
27
3x
=
3
3
x=9
8
(1)
(2)
Substitute x = 9 into (1) and solve for y
...
Substitute these values into equations (1) and (2) to make
sure they are both satisfied
...
x + y = 9 + 11
= 20
(1)
Exercises V
Solve the systems
...
(a) x − 2y = 6
3x + y = 25
(b) 2x + 3y = 17
−x − y = −4
9
(2)
Problem Set
1
...
(a) (2 + 3) × 52 − (15 − 20 ÷ 5)2
(b) ((−2 + 23 ) × 4 − (10 ÷ 2)2 )2
(c) (23 × 32 − 92 ) ÷ 3
(d) 2(30 − (10 − (4 + 12 ÷ 4))3 ) − 6
2
...
If a = b = c = d and a + b + c + d = 16, what is the value of a × b × c × d?
4
...
During a
“Numbers Round” you draw six numbers,
75 2 5 6 1 4
and a target
...
You can only use each
number once, but you do not have to use all of your numbers
...
If is an operation defined as p q = p2 + 3pq − 2q + 1, what is the value of 7 5?
6
...
(a) 3 × 57
(b) 5 × 371
1
(c) × 1256
2
(d) 23 × 32
(e) 17 × 142
7
...
Show all of your work
...
The product of 2, 4, 6, and x is equal to its sum
...
Clark scored a total of 36 points in his basketball team’s first four games
...
How many points did he score in the fourth game?
10
...
Expand and simplify the expressions
...
*Each of the five tins below contains either coffee, cocoa, or powdered milk
...
No three tins contain the same item
...
(a) 4
(c) −3
(b) 1
(d) 0
2
...
256
4
...
See solutions
...
145
6
...
(a) x = −5
8
...
13 points
10
...
(a) a2 + 2ab + b2
(b) a2 − 2ab + b2
12
Title: Algebraic order of operation notes
Description: Algebraic number theory is a branch of mathematics that studies the properties of algebraic numbers, which are complex numbers that are roots of polynomials with integer coefficients. Algebraic numbers include rational numbers, irrational numbers such as square roots, cube roots, and other radicals, and complex numbers that are algebraic extensions of the rationals. Algebraic number theory studies the algebraic properties of these numbers, such as their algebraic degree, which is the degree of the polynomial that they satisfy, and their algebraic conjugates, which are the other roots of the polynomial.
Description: Algebraic number theory is a branch of mathematics that studies the properties of algebraic numbers, which are complex numbers that are roots of polynomials with integer coefficients. Algebraic numbers include rational numbers, irrational numbers such as square roots, cube roots, and other radicals, and complex numbers that are algebraic extensions of the rationals. Algebraic number theory studies the algebraic properties of these numbers, such as their algebraic degree, which is the degree of the polynomial that they satisfy, and their algebraic conjugates, which are the other roots of the polynomial.