In mathematics one start from definitions and axioms and derive useful truth about those mathematical object. The definition of distance in mathematics serves as the perfect oppurtunity to concretize this system:
Pick any two numbers, x y their distance d(x,y) equals absolute x - y
ABsolute value on the other hand is defined as give definition piecwise of absolute value
If you have 40 on the first midterm 60 on the second midterm exam. You improved by +10 mark. The absolute value of your increase is 10. If you get 50 on the final your grade changed by -10 marks but the absolute value of your change in grade is absolute -10 = 10. Absolute strips of
Result number 1 : absolute value is non negative. As we can see from the definition. It will absolute value takes a negative number it does a procedure that always change it to positive
If x = 0 absolute value must be zero. and vice versa
By how we defined it if x = 0 aboslute value turns into 0. Think of it like zero is neutral. BUt absolute value only cares about those negative signs. it elimanates it.
on the other hand if we see absolute x = 0 what can we say about the original x?
There are three types of number positive, negative and zero? ANd each is exactly of that categeroy. So If you think of the absolute value as a machine, if it spits a zero, could the number he received be a positive? The absolute value machine does noting to the positive number it takes, So if you enter positive you can come out negative. If you enter negative you come out positive. The only way to come out zero is to enter zero. If anything is positive consider. If a zero comes out of that aboslute value machine you know what entered.
Abslute value of x smaller or equal to y gives a new interval with a lower bound of -y and upper of y.
You’re stuck in a desert. At point x Draw the line. Given the heat you can only survives absolute value kilometer without dying. If you go straight ahead we consider that you walked . If you go back to where you came from you walked - backward. If I give you . Then can you reach the any of the oasis?
A picture showing the situation
Link it to the stateement
Triangle inequality
From adrenaline you can muster the strength to walk 2 more steps after you walk your 2 meters. If you walk your 3 steps forward then 2 steps forward In the postive direction you are guaranteed to reach the oasis. If someone on the other end walk 3 steps forward but used their 2 steps to go back again can they reach? the oasis? That’s what is encapuslated in the triangel property. Walking positively will always lend you the farthest right? on the number line. Compared to adding number
Now how to draw a triangle from 2 sides lenght 2 and 3 remove them. If you place them one after the other can you create a triangle?
Show a picture that show them that you can draw a third sides to form a triangle?
What about if create an angle from the line now you can create tirangel ?
Show a lot o f bent angle creating different triangle.
All those bents are shorter compared to our third sides showing a picture of that. SO 2 sides of a triangle must always be greater or equal to the third sides in lenght. Notice that we use distance and lenght interchangeably that’s wny this theorem is called triangle inequality
If you gain 100000 you end up with positive 100000. But if you spend that 100000 three times you balance end up at -100000 . But the absolute value in both of those multiplication is 300000. Justifying the stateement Show that statement about mutiplying absolute value
Absolute value of distance
What’s distance as a computation that gives you the lenght between any point on the number line