- WELCOME TO A PRESENTATION ON REAL NUMBERS.
THE GOALS OF THIS VIDEO ARE TO DETERMINE THE DIFFERENT TYPES
OF REAL NUMBERS, COMPARE IN ORDER REAL NUMBERS,
AND ALSO TO DETERMINE THE OPPOSITE VALUE OF REAL NUMBERS.
A SET IS A COLLECTION OF OBJECTS.
THE SET CONTAINING "A," B AND C IS USUALLY WRITTEN WITH BRACES,
"A" COMMA, B COMMA, C.
THE SET OF REAL NUMBERS IS MADE UP OF ALL RATIONAL
AND IRRATIONAL NUMBERS, WHICH WE'LL DEFINE SHORTLY.
YOU CAN THINK OF THE SET OF REAL NUMBERS AS ANY NUMBER
THAT WOULD APPEAR ON THE NUMBER LINE.
SO THE REAL NUMBERS ARE THE NUMBERS
THAT CAN BE WRITTEN IN DECIMAL NOTATION,
INCLUDING THOSE THAT REQUIRE AN INFINITE DECIMAL EXPANSION.
THE SET OF REAL NUMBERS INCLUDES ALL INTEGERS,
POSITIVE AND NEGATIVE, ALL FRACTIONS
AND ALL THE IRRATIONAL NUMBERS, THOSE WHOSE DECIMAL EXPANSIONS
NEVER REPEAT.
THIS TREE OF REAL NUMBERS GIVES YOU A PRETTY GOOD IDEA
OF THE DIFFERENT TYPES OF REAL NUMBERS.
SO, AGAIN, REAL NUMBERS ARE THE SET OF NUMBERS
CORRESPONDING TO ANY POINT ON THE NUMBER LINE.
HERE IS A VARIETY OF EXAMPLES.
ALL REAL NUMBERS ARE EITHER RATIONAL OR IRRATIONAL NUMBERS.
THE DEFINITION OF AN IRRATIONAL NUMBER
IS THE SET OF ALL NUMBERS, "A" DIVIDED BY B
WHERE "A" AND B ARE INTEGERS.
BUT IT'S PROBABLY EASIER TO THINK OF A RATIONAL NUMBER
AS A NUMBER THAT IN DECIMAL FORM,
IT EITHER TERMINATES OR REPEATS WITH A PATTERN.
SO, FOR EXAMPLE, ALL OF THESE, IF I PUT THEM IN DECIMAL FORM,
WOULD EITHER TERMINATE OR HAVE A DECIMAL THAT REPEATS.
SO, FOR EXAMPLE, 5/8 IS 0.625.
IT TERMINATES, AND IF WE CONSIDERED A NUMBER
LIKE 1/3 IN DECIMAL FORM, THIS WOULD BE 0.333 REPEATING.
SO IT WOULD ALSO BE A RATIONAL NUMBER.
HOWEVER, IF WE LOOK AT IRRATIONAL NUMBERS,
IN DECIMAL FORM, THESE NEITHER TERMINATE NOR REPEAT.
SO THEY GO ON FOREVER WITHOUT A PATTERN.
SO, IF YOU TAKE THE SQUARE ROOT OF 19 OR YOU LOOK AT THE DIGITS
OF PI, IT GOES ON FOREVER, BUT THERE'S NO PATTERN
IN THE DECIMAL PART OF THE NUMBER.
NOW, IF WE TAKE A LOOK AT THE SET OF RATIONAL NUMBERS,
THESE CAN BE BROKEN DOWN INTO INTEGERS
AND THEN RATIONAL NUMBERS THAT ARE NOT INTEGERS.
SO, HERE IS THE SET OF INTEGERS, AND WE CAN SEE THESE VALUES
FIT INTO THIS SET.
AND RATIONAL NUMBERS THAT ARE NOT INTEGERS
WOULD CONSIST OF FRACTIONS AND DECIMALS.
INTEGERS CAN BE BROKEN DOWN INTO NEGATIVE INTEGERS
AND WHOLE NUMBERS.
I THINK NEGATIVE INTEGERS ARE PRETTY OBVIOUS.
WHOLE NUMBERS START AT 0 AND INCREASE BY 1.
SO THESE WOULD BE OUR WHOLE NUMBERS
FROM OUR ORIGINAL SET.
AND THEN WE CAN BREAK DOWN WHOLE NUMBERS
INTO 0 AND POSITIVE INTEGERS.
THE ONLY DIFFERENCE BETWEEN A POSITIVE INTEGER
AND A WHOLE NUMBER IS 0 IS NOT CONSIDERED A POSITIVE INTEGER.
SOMETIMES THESE ARE CALLED NATURAL NUMBERS
OR COUNTING NUMBERS.
SO YOU MAY WANT TO PAUSE THE VIDEO
AND LOOK AT THIS A LITTLE BIT CLOSER,
BUT I THINK ORGANIZING IT IN A TREE DIAGRAM IS VERY HELPFUL.
OKAY. LET'S GO AHEAD AND TAKE A LOOK AT COMPARING REAL NUMBERS
USING INEQUALITY SYMBOLS.
WE HAVE THE < SYMBOL, < OR = SYMBOL,
>, > OR = SYMBOL.
TO COMPARE NUMBERS, IT IS OFTEN USEFUL TO CONVERT
TO DECIMAL NOTATION AND/OR PLOT ON THE NUMBER LINE.
SMALLER NUMBERS ARE ALWAYS TO THE LEFT OF LARGER NUMBERS
ON THE NUMBER LINE.
SO IF WE WANT TO COMPARE THESE TWO NUMBERS,
-3 AND -5, AND WE CAN'T TELL JUST BY LOOKING WHICH ONE
IS SMALLER OR LARGER, WE COULD JUST MAKE A NUMBER LINE,
AND WE WOULD SEE -3 OCCUR SOMEWHERE IN HERE AND -5 HERE.
SINCE -5 IS TO THE LEFT OF -3, WE CAN SAY -5 < -3
OR WE CAN SAY -3 > -5.
EITHER WAY, -3 IS THE LARGER OF THE TWO.
SO TO COMPARE USING THESE TWO INEQUALITY SYMBOLS,
WE WOULD SAY -3 > -5.
NEXT, COMPARING 2/5 AND 4/7 MAY BE DIFFICULT
SINCE WE EITHER NEED TO HAVE A COMMON DENOMINATOR
OR CONVERT TO DECIMAL FORM.
SO IF WE CONVERT 2/5 TO DECIMAL FORM,
WE NEED TO TAKE 2 AND DIVIDE BY 5,
AND THAT WOULD GIVE US 0.4.
AND IF WE TAKE 4 AND DIVIDE BY 7,
WE'LL GET APPROXIMATELY .57.
AGAIN, IN DECIMAL FORM IT'S EASIER TO SEE
THAT 4/7 IS LARGER THAN 2/5.
THEREFORE, WE WOULD READ THIS 2/5 < 4/7.
THE SAME THING WITH NEGATIVE SQUARE ROOT 8
AND NEGATIVE SQUARE ROOT 7.
LET'S CONVERT THOSE TO DECIMAL FORM.
SO LET'S GO AHEAD AND USE OUR CALCULATOR,
AND WE'LL TYPE IN -2nd X SQUARED 8,
CLOSE THE PARENTHESES, AND THEN WE'LL ALSO FIND
THE DECIMAL VALUE OF NEGATIVE SQUARE ROOT 7.
SO LET'S GO AHEAD AND JUST PLOT -2.8 AND -2.6
ON THE NUMBER LINE.
SO THE QUESTION REALLY BECOMES, "WHICH WOULD COME FIRST,
-2.6 OR -2.8?"
AND IT WOULD BE -2.6, -2.8.
AGAIN, WE KNOW THAT BECAUSE -2 MIGHT BE HERE
AND -3 MIGHT BE HERE.
-2.8 IS CLOSER TO -3.
THEREFORE, -2.6 IS LARGER AND THIS CAME FROM
NEGATIVE SQUARE ROOT 7.
SO WE CAN STATE THAT NEGATIVE SQUARE ROOT 8
< NEGATIVE SQUARE ROOT 7.
AND LASTLY, THIS ONE'S PRETTY STRAIGHTFORWARD.
A POSITIVE NUMBER IS ALWAYS LARGER THAN A NEGATIVE NUMBER.
SO WE KNOW THAT -4.6 < 5.3.
SO I HIGHLY RECOMMEND CONVERTING TO DECIMAL FORM
AND THEN USING THE NUMBER LINE TO DETERMINE WHICH NUMBER
IS < OR > THE OTHER.
NOW WE HAVE TRUE AND FALSE.
-3 IS < OR = -3.
THIS WOULD BE TRUE, BECAUSE -3 = -3,
AND = IS PART OF THIS INEQUALITY SYMBOL.
THEREFORE, THIS IS TRUE.
NEXT, AGAIN, WE'LL CONVERT THIS TO DECIMAL NOTATION.
SO WE'D HAVE -0.75 COMPARED TO -1.
-0.75 IS > -1.
SO, OF COURSE, IT WOULD ALSO BE > OR =.
AGAIN, THIS IS TRUE.
AND LASTLY, 0 < -3.
AGAIN, WE CAN ALWAYS GO BACK TO THE NUMBER LINE.
IS 0 < 3?
NO, BECAUSE 0 IS TO THE RIGHT OF -3.
SO THIS WOULD BE FALSE.
OKAY. LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE TOPIC.
LET'S TALK ABOUT ABSOLUTE VALUE.
WHEN WE WRITE "A" BETWEEN TWO VERTICAL BARS,
THIS READS "THE ABSOLUTE VALUE OF A,"
WHICH REPRESENTS THE NUMBER OF UNITS OR DISTANCE
THAT "A" IS FROM 0.
SO, SINCE DISTANCE IS ALWAYS POSITIVE,
WHEN WE FIND AN ABSOLUTE VALUE, IT WILL ALWAYS BE POSITIVE.
SO, WHEN WE GRAPH Y = ABSOLUTE VALUE OF X,
WE CAN SEE THAT ALL OF THE POINTS
EXCEPT THE POINT (0,0) ARE ABOVE THE X AXIS.
THEREFORE, ALL THE Y VALUES ARE POSITIVE OR COULD BE 0.
SO, THE ABSOLUTE VALUE OF -8, HOW FAR IS -8 FROM 0?
IT'S 8 UNITS FROM 0.
AND THE ABSOLUTE VALUE OF 7.5 IS 7.5.
AND THE ABSOLUTE VALUE OF 0 IS 0.
SO THE ONLY THING THAT I'VE SEEN THAT SOMETIMES CAUSES AN ISSUE
IF YOU'RE ASKED TO EVALUATE THE NEGATIVE ABSOLUTE VALUE
OF A NEGATIVE NUMBER, REMEMBER, YOU CAN THINK OF THIS
AS A NEGATIVE OR THE OPPOSITE.
SO, THE OPPOSITE OF THE ABSOLUTE VALUE OF -3,
WELL, WE KNOW THIS WOULD BE +3,
BUT THE OPPOSITE OF +3 IS GOING TO BE A -3.
SO, THERE'S A NEGATIVE SIGN OUTSIDE OF THE ABSOLUTE VALUE
THAT WOULD RETURN A NEGATIVE VALUE.
OKAY, I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING AND HAVE A GOOD DAY.