- NOW, WE'RE GOING TO REVIEW DIFFERENT SETS OF REAL NUMBERS.
REAL NUMBERS, IDENTIFIED BY CAPITAL R
ARE NUMBERS THAT WOULD APPEAR ON THE NUMBER LINE.
AND ALL REAL NUMBERS ARE EITHER RATIONAL NUMBERS
OR IRRATIONAL NUMBERS, BUT BEFORE WE DEFINE
THESE TWO SETS OF NUMBERS, I'D LIKE TO START
WITH THE NATURAL NUMBERS OR COUNTING NUMBERS.
THEN WE'LL WORK OUR WAY UP.
SO, NATURAL NUMBERS ARE COUNTING NUMBERS
IDENTIFIED BY CAPITAL N OF THE NUMBERS 1, 2, 3, 4, 5, 6,
AND SO ON.
SO, LOOKING AT OUR VENN DIAGRAM HERE,
THE NATURAL NUMBERS WOULD BE THESE NUMBERS
IN THIS SET HERE, AGAIN, IDENTIFIED BY CAPITAL N.
NEXT, THE WHOLE NUMBERS INCLUDE ALL OF THE NATURAL NUMBERS
AND THE NUMBER 0.
SO, LOOKING AT THE VENN DIAGRAM AGAIN,
THE CAPITAL W REPRESENTS WHOLE NUMBERS WHICH INCLUDE 0
AS WELL AS ALL OF THE NATURAL NUMBERS.
THE INTEGERS IDENTIFIED BY CAPITAL Z OF THE NUMBERS
THAT BELONG TO THIS SET HERE, NOTICE HOW IT INCLUDES
THE NEGATIVE NATURAL NUMBERS, 0, AND THE NATURAL NUMBERS.
SO, AGAIN, LOOKING AT OUR VENN DIAGRAM,
HERE'S Z, HERE ARE THE NEGATIVE NATURAL NUMBERS, 0,
AND THE NATURAL NUMBERS.
ALL OF THESE ARE INCLUDED IN THE SET OF INTEGERS,
WHICH NOW BRINGS US TO THE SET OF RATIONAL
AND IRRATIONAL NUMBERS.
SO, WE CAN SEE RIGHT AWAY THAT INTEGERS, WHOLE NUMBERS
AND NATURAL NUMBERS ARE WITHIN THE SET Q,
WHICH ARE RATIONAL NUMBERS.
RATIONAL NUMBERS CAN BE WRITTEN IN THE FORM
OF "A" DIVIDED BY B WITH "A" AND B AS INTEGERS.
OF COURSE, B CAN'T = 0.
SO WE HAVEN'T INCLUDED FRACTIONS YET, BUT FRACTIONS
LIKE 2/3 AND -2/5 WOULD ALSO BE RATIONAL NUMBERS,
EVEN THOUGH THEY DON'T BELONG TO THE SET OF INTEGERS,
WHOLE NUMBERS OR NATURAL NUMBERS.
IN DECIMAL FORM, RATIONAL NUMBERS WOULD EITHER
TERMINATE OR REPEAT.
SO, FOR EXAMPLE, 0.4 TERMINATES AND THEREFORE IS RATIONAL,
AND 0.3 REPEATS, BECAUSE THE 3 REPEATS
AND THEREFORE IS ALSO RATIONAL.
SO A LOT OF TIMES WHEN TRYING TO DETERMINE
IF A NUMBER IS RATIONAL, IT CAN BE HELPFUL
TO CONVERT IT TO A DECIMAL.
AND THEN IRRATIONAL NUMBERS ARE NUMBERS THAT,
IN DECIMAL FORM, DO NOT TERMINATE AND DO NOT REPEAT.
SO, FOR EXAMPLE, SQUARE ROOT 5
WOULD BE SOME NON-TERMINATING, NON-REPEATING DECIMAL,
AND OF COURSE, SO IS PI.
SO LET'S TAKE A LOOK AT A FEW EXAMPLES.
WE WANT TO DETERMINE WHICH SET OR SETS
OF REAL NUMBERS EACH NUMBER WOULD BELONG TO.
SO, NUMBER ONE, WE HAVE 0.
LET'S GO BACK AND TAKE A LOOK AT OUR VENN DIAGRAM.
NOTICE THAT 0 IS HERE, WHICH MEANS IT'S A WHOLE NUMBER.
THEN WE CAN WORK OUR WAY OUT.
Z REPRESENTS THE SET OF INTEGERS.
SO 0 IS AN INTEGER.
Q REPRESENTS THE RATIONAL NUMBERS.
SO, 0 IS ALSO RATIONAL.
THEN, OF COURSE, ALL THE NUMBERS ARE GOING TO BE REAL NUMBERS.
SO, AGAIN, IT'S GOING TO BE A WHOLE NUMBER, SO W.
IT'S GOING TO BE AN INTEGER, WHICH IS Z.
IT'S GOING TO BE RATIONAL, WHICH IS Q, AND, OF COURSE,
IT'S ALSO A REAL NUMBER.
NEXT, WE HAVE 2/3. OKAY.
AND LET'S GO BACK AND TAKE A LOOK AT OUR VENN DIAGRAM.
WELL, 2/3 IS IN THE FORM OF A/B.
SO IT'S RATIONAL, AND ALL RATIONAL NUMBERS
ARE ALSO REAL.
SO IT BELONGS TO THE SET OF RATIONAL NUMBERS
AND THE SET OF REAL NUMBERS.
NEXT WE HAVE THE SQUARE ROOT OF 8.
LET'S GO AHEAD AND CONVERT THIS ONE TO A DECIMAL.
SO, SECOND X SQUARED, WHICH BRINGS UP THE SQUARE ROOT,
AND THEN WE HAVE 8.
NOTICE HOW THIS IS A NON-TERMINATING,
NON-REPEATING DECIMAL.
SO LET'S GO BACK AND TAKE A LOOK AT OUR VENN DIAGRAM
ONE MORE TIME.
IT'S A NON-REPEATING, NON-TERMINATING DECIMAL,
WHICH MEANS IT'LL BELONG TO THIS SET UP HERE,
WHICH IS THE SET OF IRRATIONAL NUMBERS.
REMEMBER, IN DECIMAL FORM IT DOES NOT TERMINATE
AND DOES NOT REPEAT.
AND ALL IRRATIONAL NUMBERS ARE ALSO REAL,
SO IT'S IRRATIONAL AND IT'S ALSO REAL.
NOW, NEGATIVE SQUARE ROOT 36, WE MIGHT THINK
THAT THIS IS ALSO IRRATIONAL, BUT REMEMBER
THE SQUARE ROOT OF 36 IS 6, SO THIS SIMPLIFIES TO -6.
SO WE WANT TO USE -6 TO DETERMINE THE SETS
THIS NUMBER BELONGS TO.
WELL, -6 WOULD BE AN INTEGER, WHICH MEANS IT'S ALSO
A RATIONAL, AND IT'S ALSO REAL.
IN THE LAST EXAMPLE WE HAVE PI.
LET'S TYPE PI INTO THE CALCULATOR
AND SEE WHAT IT GIVES US AS A DECIMAL.
I KNOW WE OFTEN USE 3.14 AS AN APPROXIMATION,
BUT PI ACTUALLY IS A NON-TERMINATING DECIMAL,
AND IT'S ALSO A NON-REPEATING, BECAUSE THERE'S NO PATTERN HERE.
SO THAT MEANS PI IS IRRATIONAL AND ALSO REAL.
SO IT'S I AND R.
NOW, I WANT TO GO BACK TO 2/3 JUST FOR A MINUTE.
LET'S GO AHEAD AND CONVERT THIS TO A DECIMAL TO VERIFY
THAT IT ALSO SATISFIES THE DECIMAL DEFINITION
FOR A RATIONAL NUMBER.
SO TO CONVERT 2/3 TO A DECIMAL, WE WOULD HAVE 2 DIVIDED BY 3.
NOW, THE CALCULATOR IS ROUNDING HERE.
THIS IS NON-TERMINATING, BUT NOTICE HOW THE 6 IS REPEATING.
SO WE HAVE A REPEATING NON-TERMINATING DECIMAL.
SO IF WE GO BACK AND TAKE A LOOK AT THE DEFINITION
OF A RATIONAL NUMBER, AGAIN, NOTICE HOW IT SAYS,
"IN DECIMAL FORM IT TERMINATES OR REPEATS."
AND, AGAIN, 2/3 WAS .6 REPEATING,
VERIFYING THAT IT IS A RATIONAL NUMBER AND ALSO REAL.
OKAY, SO, I HOPE YOU FOUND THESE EXAMPLES HELPFUL.
IT MAY ALSO BE HELPFUL TO TAKE A LOOK AT THESE SETS OF NUMBERS
AS A TREE DIAGRAM.
SO YOU MAY WANT TO PAUSE THE VIDEO HERE
AND REVIEW IT IN THIS FORMAT AS WELL.
HERE WE HAVE THE SET OF REAL NUMBERS,
RATIONAL AND IRRATIONAL,
INTEGERS,
WHOLE NUMBERS,
AND NATURAL NUMBERS.
OKAY, I HOPE YOU FOUND THIS HELPFUL.