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Real numbers
Collection of all rational and irrational numbers, both positive and negative, including zero.

Collection of all rational and irrational numbers, both positive and negative, including zero.
Real numbers

Integers
Counting numbers 1,2,3....together with 0 and the negative numbers 1,2,3....

Counting numbers 1,2,3....together with zero and the negative numbers 1,2,3....
Integers

Natural numbers
 All positive integers.
 Also called counting numbers.

All positive integers.
Natural numbers

Counting numbers
Natural numbers

Two consecutive negative signs
Double negative

Double negative
Two consecutive negative signs

Distributive law of multiplication
a(b + c) = ab + ac

a(b + c) = ab + ac is the _.
distributive law of multiplication.

Rational numbers
Can be expressed as a ratio of integers or as a fraction.

Can be expressed as a ratio of integers or as a fraction
Rational numbers


A fraction is reduced if _.
the numbers a and b have no common factors.

A fraction is _ the numbers a and b have no common factors.
reduced

Another way to say the fraction is reduced is to say _.
that it is in lowest terms.







General rule for adding fractions:


OR

General form for subtracting fractions

Subtract from

Reciprocal of a number x when multiplied by x yeilds _.
1

Reciprocal of any integer n (except 0) is _.

Reciprocal of is _.
n

Find the reciprocal of


General rule for division

Irrational numbers
 A number that is not rational.
 Cannot be written as a fraction of integers.

Cannot be written as a fraction of integers.
Irrational

Important class of irrational numbers
Square roots of positive integers that are not perfect squares.

Square roots of positive integers that are not perfect squares.
Important class of irrational numbers.

Integers from 2 and above are divided into two categories:

Prime number
 Not the product of two integers bigger than 1.
 Divisible by itself and 1 only.

Not the product of two integers bigger than 1.
Prime number

Divisible by itself and 1 only.
Prime number

Composite number
Integer that is equal to or greater than 2 and is not prime.

Integer that is equal to or greater than 2 and is not prime:
Composite number

Set of integers is closed under addition because _.
whenever two integers are added, the result is another integer.

Whenever two integers are added, the result is another integer, causing the set of integers to _.
be closed.

Set of integers is not closed under division, because _.
it is not always true that an integer divided by another integer yields an integer.

Set of negative integers is (close/not closed) under multiplication.
not closed

Absolute value of a number x is denoted _.
x

x denotes _.
absolute value of a number

Absolute value
Distance from the number to 0.

Distance from a number to 0.
Absolute value

 

Is every integer a rational number? Why?
Yes. Each integer can be written as a fraction with 1 in the denominator, like 4 = .

Is a rational number? Is rational? How about ?
is irrational. is rational because it is the integer of 3. is not even real, so it is certainly not rational.

List the prime numbers between 50 and 60.
53 and 59 are the only primes between 50 and 60.

Is rational?
No. is close to both and 3.14, but it is not equal to either.

Are the negative integers closed with respect to the operation of addition?
Yes. Whenever you add two negative integers, the result is another negative integer.

Is the product of two irrational numbers always irrational?
No. Consider times = 2. is irrational, but 2 is not.

Is it true that for any real numbers a, b, and c, a(b  c) = ab  ac?
Yes. Subtracting a number is equivalent to adding its negative, so the distributive law applies here: a X (b + (c) = a X b + a X (c).

Write as a single fraction.

Write as a single fraction.

 is approximately 3.14159 (is the smallest),
 and is approximately .571429
 is approximately 1.7320508,
 = 3, and
 = 10 (is the largest)

Are 3  1 and 1  3 equal?
Yes. 3  1 = 2 = 2. 1  3 = 2 = 2.

Is it true that for real numbers a and b, a + b = a + b? Provide an example where this is false.
No. When the numbers have opposite signs, it is false. Consider a = 6 and b = 4. 6 + (4) = 6  4 = 2 = 2, while 6 + 4 = 6 + 4 = 10. 2 10, so the statement is not true for all a and b.

Exponents
Multiplying a number by itself several times

Multiplying a number by itself several times
exponent

First law of exponents
a^{n}a^{m} = a^{n+m}
Adding the powers of exponents

The law of adding powers of exponents
First law of exponents

Second law of exponents
(a^{b})^{c} = a^{bc}
Multiplying the powers of exponents

Law of multiplying the powers of exponents
Second law of exponents

^{}Third law of exponents
a^{c}b^{c} = (ab)^{c}
Exponent roots are paired off

^{}Law of pairing of the exponent roots
Third law of exponents

Last rule of exponents
 Involves fractions
 = a^{nm}

Law that involves fractions
Last law of exponents

Variable
Letter that represents some unknown or undetermined number.

Letter that represents some unknown or undetermined number.
Variable

(4ab)^{3}3a^{2}b =
4^{3}a^{3}b^{3}3a^{2}b^{ }= 4^{3}3a^{3}a^{2}b^{3}b = 64 X 3a^{5}b^{4} = 192a^{5}b^{4}

=
= 2 ^{53}3 ^{23}^{ }= 2 ^{2}3 ^{1} =

=

=
3 ^{2}x ^{2}(3x ^{3}) ^{1} = 3 ^{2}x ^{2}3 ^{1}x ^{3} = 3 ^{1}x ^{5} = 3x ^{5}
OR
= = 3x ^{5}


=
3^{2}





=
x^{mn}

m
=

n
=
 n






2^{5}2^{7} =
2^{12} = 4096

^{}(3^{2})^{3} =
3^{6} = 729

^{ =}
4^{2 }= 16

=
(2^{1})^{3} = 2^{3} = 8



Is there a difference between 2x^{4} and (2x)^{4}?
Yes. In 2x^{4}, only the x is being raised to the 4th power. (2x)^{4}, on the other hand, is 2^{4}x^{4}= 16x^{4}.

=
3^{51}x^{52}y^{21} = 3^{4}x^{3}y= 81x^{3}y

(2xy)^{2}(2x^{2}y)^{2} =
^{}2^{2}x^{2}y^{2}2^{2}x^{4}y^{2} = 2^{4}x^{6}y^{4} = 16x^{6}y^{4}

=
Only x's cancel.

=
(r^{2}s)^{4}(s^{2}r^{2}) = r^{8}s^{4}s^{2}r^{2} = r^{10}s^{2}

=


