Real Numbers - BakerMath.org
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Transcript Real Numbers - BakerMath.org
Real Numbers
1
2
3
Definition
Properties
Examples
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Definition
2
Real Numbers include:
Integers
-3,-2,-1,0,1,2,3
Rational
Numbers
Decimals
that can be represented in fraction form that are
either terminating or non-terminating and repeating
5/4 = 1.25
177/55 = 3.2181818…
1/3 = .33333…
Irrational
Numbers
Non-terminating
and non-repeating decimals
Π = 3.14159…, √2 = 1.41421…
Properties
3
Addition is commutative
a
+b=b+a
Order does not matter
Addition is associative
a
+ (b + c) = (a +b) + c
Grouping does not matter
0 is the additive identity
a
+0=a
Adding 0 yields the same number
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Properties (Cont.)
4
-a is the additive inverse (negative) of a
a
+ (-a) = 0, 12+(-12)=0
Adding a number and it’s inverse gives 0
Multiplication is commutative
ab
= ba, 3*4=4*3=12
Order of multiplication does not change the result
1 is the multiplicative identity
a
*1=a
Multiplying 1 yields the same number
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Properties (Cont.)
5
If a ≠ 0, 1/a is the multiplicative inverse (reciprocal)
of a
a(1/a) = 1, 3(1/3)=1
Multiplying a non-zero number by its reciprocal yields 1
Multiplication is distributive over addition
a(b + c) = ab + ac
(a + b)c = ac + bc
Multiplying a number and a sum of two numbers is the same
as multiplying each of the two numbers by the multiplier and
then adding the products
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Properties (Cont.)
6
Trichotomy Law
Definition of Absolute Value
If a ≥ 0, then |a|=a
If a <0, then |a|=-(a)
Distance on a number line
If a and b are real numbers, then exactly one of the following is
true: a=b, a<b, a>b
d(A, B) = |B-A|
Law of the signs
If a and b both have the same sign, then ab and a/b are positive
If a and b have different signs, then ab and a/b are negative
Examples
7
If p, q, r, and s denote real numbers, show that
(p+q)(r+s)=pr+ps+qr+qs
If x>0, and y<0, determine the sign of x/y + y/x
(p+q)(r+s)
=p(r+s)+q(r+s)
=(pr+ps)+(qr+qs)
= pr+ps+qr+qs
Since only y is negative, both x\y and y/x will be negative numbers
A negative number increased by another negative number will yield a
“more” negative number
If x<1, rewrite |x-1| without using the absolute value symbol
If x<1, then x-1<0 (negative)
By part 2 of the definition of absolute value, |x-1|=-(x-1)=-x+1 or 1-x
Examples
8
Let A, B, C, and D have coordinates -5, -3, 1, and 6
respectively. Find d(B,D)/\.
d(B,D)
= d(-3,6)
=|6-(-3)|
=|6+3|
=|9|
=9
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Guided Practice
9
Do Problems on page 16, 1-40
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