Geometry: Week 1
Why can we not divide zero from both sides? 2 reasons
-We aren't allowed to divide by zero. 1/0 doesn't make any sense. 1/0 = undefined -Creates conflict in an equation 0 x 1 = 0 x 2 We cannot cancel zero from both sides ----- 0/0 = indeterminate
Different Relationships eg. Elephant = Animal 1) 2) 3)
1) Animal is not always an Elephant - No. 2) Animal is not always an Elephant but Elephant is always an Animal - No. 3) Can't apply because Animal is not always Elephant so how can it be set equal to something else, etc?
Why do we need these relationships?
A lot of time, we apply these when solving problems e.g. 2x+1 = 7 so 7=2x+1
Reflexive Property
A=A
Transitive
A=B and B=C so A=C
Symmetric Property
A=B so B=A
Name the three properties of equations
Reflexive, Symmetric, Transitive
REMEMBER!!!!!!
We CANNOT divide zero from both sides!!! 0 erases everything.
Algebraic Properties of Equality 1
We can add or subtract equivalent value from both sides of the equation, simultaneously. If a=b, then a+c = b+c
Algebraic Properties of Equality 2
We can multiply or divide the same non-zero value from both sides of the equation simultaneously. If a=b, then 2a=2b or 3a=3b or ka=kb (k =/ 0)
We want a --------- side and a ----------- side
a constant and a variable side
Equations
a relationship between two expressions that the 2 expressions are of the same value
Prove by induction
hard to exhaust all experience
In equations the goal is to...
isolate the variable (x)
What is induction in math?
proving that a statement is true for every natural number, that is infinitely many cases
Multiplication
repeated addition
We can we not always divide by a variable
this is because if we don't know whether it is equal to zero or not, we cant because it is improper to divide by zero
If we can't prove by induction we can prove by using
using a property or something we know to prove
To prove that something is true...
we have to argue and present reasons e.g. if sunny then temp rises to prove we need to collect data
To prove that something is false...
you only need to show a counter example. e.g. it is raining outside, we need to check outside of a window
