Post on 02-May-2017
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Real NumbersReal NumbersRational and IrrationalRational and Irrational
Let’s look at the relationships between number sets. Notice rational and
irrational numbers make up the larger number set known as Real Numbers
A number represents the value or quantity of something… Like how much money you have.. Or how many marbles
you have… Or how tall you are.As you may remember from earlier grades
there are different types of numbers.
A number line - is an infinitely long line whose points match up with the real
number system.
Here are the rational numbers represented on a number line.
IntegersIntegersThe coldest temperature on record in The coldest temperature on record in the U.S. is -80° F, recorded in 1971 in the U.S. is -80° F, recorded in 1971 in
AlaskaAlaska
Integers are used to represent real-world quantities such as temperatures, miles per hour, making withdrawals from your bank account, and other quantities. When you
know how to perform operations with integers, you can solve equations and
problems involving integers.
By using integers, you can express elevations above, below, and at sea level.
Sea level has an elevation of 0 feet. Badwater Basin in Utah is -282 below sea level, and Clingman’s Dome in the Great Smokey Mountains is +6,643 above sea
level.
If you remember, the whole numbers are the counting numbers and zero:
0, 1, 2, 3,…Integers - the set of all whole numbers and their opposites. This means all the positive integers and all the negative
integers together.
Opposites – two numbers that are equal distance from zero on a number line; also
called additive inverse.The additive inverse property states that if you add two opposites together their sum
is 0-3 + 3 = 0
Integers increase in value as you move to the right along a number line. They
decrease in value as you move to the left. Remember to order numbers we use the
symbol < means “less than,” and the symbol > means “is greater than.”
A number’s absolute value - is it’s distance from 0 on a number line. Since distance can
never be negative, absolute values are always positive. The symbol || represents the absolute value of a number. This symbol is
read as “the absolute value of.” For example |-3| = 3.
Finding absolute value using a number line is very simple. You just need to know the
distance the number is from zero. |5| = 5, |-6| = 6
Lesson Quiz
Compare, Use <, >, or =.1) -32 □ 322) 26 □ |-26|3) -8 □ -124) Graph the numbers -2,
3, -4, 5. and -1 on a number line. Then list the numbers in order from least to greatest.
5) The coldest temperature ever recorded east of the Mississippi is fifty-four degrees below zero in Danbury, Wisconsin, on January 24, 1922. Write the temperature as an integer.
Integer OperationsInteger OperationsRules for Integer OperationsRules for Integer Operations
Adding IntegersWhen we add numbers with the same signs,
1) add the absolute values, and2) write the sum (the answer) with the sign of the
numbers.When you add numbers with different signs,
1) subtract the absolute values, and2) write the difference (the answer) with the sign of the number having the larger absolute value.
Try the following problems
1) -9 + (-7) = -16
2) -20 + 15 = -5
3) (+3) + (+5) = +8
4) -9 + 6 = -3
5) (-21) + 21 = 0
6) (-23) + (-7) = -30
Subtracting IntegersYou subtract integers by adding its
opposite.9 – (-3)
9 + (+3) = +12
-7 – (-5)-7 + (+5) = -2
Try the following problems
1) -5 – 4 = -5 + (-4) = -9
2) 3 – (+5) =3 + (-5) = -2
3) -25 – (+25) =-25 + (-25) = -50
4) 9 – 3 =9 + (-3) = +6
5) -10 – (-15) =-10 + (+15) = +5
Multiplying and Dividing IntegersIf the signs are the same,
the answer is positive.
If the signs are different,the answer is negative.
Try the following problems
Think of multiplication as repeated addition.3 · 2 = 2 + 2 + 2 = 6 and 3 · (-2) = (-2) + (-2) + (-
2) = -6
1) 3 · (-3) = Remember multiplication is fast adding
= 3 · (-3) = (-3) + (-3) + (-3) = -9
2) -4 · 2 = Remember multiplication is fast adding
= -4 · 2 = (-4) + (-4) = -8
Dividing Integers
Multiplication and division are inverse operations. They “undo” each other. You can use this fact to discover the rules for division of integers.4 · (-2) = -8 -4 · (-2) = 8-8 ÷ (-2) = 4 8 ÷ (-2) = -4
same sign positive different signs negativeThe rule for division is like the rule for multiplication.
Try the following problems
1) 72 ÷ (-9) 72 ÷ (-9) Think: 72 ÷ 9 = 8
-8The signs are different, so the quotient is negative.
2) -144 ÷ 12 -144 ÷ 12 Think: 144 12 = 12
-12 The signs are different, so the quotient is negative.
3) -100 ÷ (-5) Think: 100 ÷ 5 = 20
-100 ÷ (-5) The signs are the same, so the quotient is
positive.
Lesson Quiz
Find the sum or difference1) -7 + (-6) =2) -15 + 24 + (-9) =Evaluate x + y for x = -
2 and y = -153) 3 – 9 =4) -3 – (-5) =Evaluate x – y + z for x = -4, y = 5, and z = -
10
Find the product or quotient1) -8 · 12 =2) -3 · 5 · (-2) =3) -75 ÷ 5 =4) -110 ÷ (-2) =5) The temperature in
Bar Harbor, Maine, was -3 F. During the night, it dropped to be four times as cold. What was the temperature then?
Rational NumbersRational NumbersFractions and DecimalsFractions and Decimals
Rational numbers – numbers that can be written in the form a/b (fractions), with
integers for numerators and denominators.
Integers and certain decimals are rational numbers because they can be written as
fractions. a 1 2 3 4 5 … b 1 1/ 1 2/ 1 3/ 1 4/ 1 5/ 1 … 2 1/ 2 2/ 2 3/ 2 4/ 2 5/ 2 … 3 1/ 3 2/ 3 3/ 3 4/ 3 5/ 3 … 4 1/ 4 2/ 4 3/ 4 4/ 4 5/ 4 … 5 1/ 5 2/ 5 3/ 3 4/ 5 5/ 5 … …
Remember you can simplify a fraction into a decimal by dividing the denominator into the numerator, or you can reduce a decimal by
placing the decimal equivalent over the appropriate place value.
O.625 = 625/1000 = 5/8
Hint: When given a rational number in decimal form (such as 2.3456) and asked to
write it as a fraction, it is often helpful to “say” the decimal out loud using the place
values to help form the fraction.2 . 3 4 5 6 o a t h t ten- n n e u h t e d n n o h s t d u o h r s u s e a s d n a T d n h t d s h t s h s
Write each rational number as a fraction:
Rational number I n decimal f orm
Rational number I n f ractional f orm
0.3 3/ 10 0.007 7/ 1000 -5.9 -59/ 10
Hint: When checking to see which fraction is larger, change the fractions to decimals by dividing and comparing their decimal
values.
Which of the given numbers is greater?
Using f ull calculator display to compare the numbers
2/ 3 and 1/ 4 .6666666667 > .25 -7/ 3 and – 11/ 3 -2.333333333 > -3.666666667
Examples of rational numbers are:
6 or 6/1 can also be written as 6.0-2 or -2/1 can also be written as -2.0½can also be written as 0.5-5/4 can also be written as -1.252/3 can also be written as .662/3 can also be written as 0.666666…21/55 can also be written as 0.38181818…53/83 can also be written as 0.62855421687…
the decimals will repeat after 41 digits
Examples: Write each rational number as a fraction:
1) 0.3
2) 0.007
3) -5.9
4) 0.45
Since Real Numbers are both rational and
irrational ordering them on a number line can be difficult if you don’t pay attention to the details.As you can see from the
example at the left, there are rational and irrational
numbers placed at the appropriate location on
the number line.This is called ordering
real numbers.
Irrational numbersIrrational numbers√√2 = 1.414213562…2 = 1.414213562…
no perfect squares hereno perfect squares here
Irrational number – a number that cannot be expressed as a ratio of two integers (fraction) or as a repeating or
terminating decimal.
• An irrational number cannot be expressed as a fraction.
• Irrational numbers cannot be represented as terminating or repeating decimals.
• Irrational numbers are non-terminating, non-repeating decimals.
Below are three irrational numbers.Decimal representations of each of these are
nonrepeating and nonterminating
10101001000.0
Examples of irrational numbers
are:
= 3.141592654…√2 =
1.414213562…0.12122122212 √7, √5, √3, √11,
343√Non-perfect squares
are irrational numbers
Note:The √ of perfect
squares are rational numbers.
√25 = 5 √16 = 4 √81 = 9Remember: Rational numbers when divided
will produce terminating or repeating decimals.
NOTE:Many students think that
is a terminating decimal, 3.14, but it is not. Yes, certain math problems ask you to use as 3.14, but that problem is rounding the value of to make your calculations easier. It is actually an infinite decimal and is an irrational number.
There are many numbers on a real
number line that are not rational. The number is not a
rational number, and it can be located on a real number line by
using geometry. The number is not equal to 22/7, which is only an approximation of
the value. The number is exactly equal to
the ratio of the circumference of a
circle to its diameter.
Enjoy your Pi