Download Types of Numbers Real Rational Irrational Integer Whole Counting

Types of Numbers
10
Real
Rational
Irrational
Integer
Whole
Counting
For each of the following numbers, circle all sets to which it belongs:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
√8
-2
51.3
0
83
0.5555…
π
0.35
-9
/2
-0.341
real
real
real
real
real
real
real
real
real
real
irrational
irrational
irrational
irrational
irrational
irrational
irrational
irrational
irrational
irrational
rational
rational
rational
rational
rational
rational
rational
rational
rational
rational
integer
integer
integer
integer
integer
integer
integer
integer
integer
integer
whole
whole
whole
whole
whole
whole
whole
whole
whole
whole
counting
counting
counting
counting
counting
counting
counting
counting
counting
counting
PROPERTIES
Commutative Property The order in which you add or multiply two numbers does not change the
sum or product.
a+b=b+a
a•b = b•a
3 + (-2) = -2 + 3
4•(-5) = -5•4
Associative Property The way you group three numbers in a sum or product does not change the
sum or product.
(a + b) + c = a + (b + c)
(-3 + 2) + 1 = -3 + (2 + 1)
(a•b)•c = a•(b•c)
(-2•7) •4 = -2•(7•4)
Identity Property The sum of zero and a number is that number. The product of a number and 1 is
that number.
a+0=a
a•1 = 1•a = a
-5 + 0 = -5
(-5)•1 = -5
Inverse Property The sum of a number and its opposite is 0.
 3 
1
 1
3
a + (-a) = -a + a = 0 -6 + 6 = 0
Property of Zero The product of a number and 0 is 0.
a•0 = 0•a = 0 -3•0=0
Property of -1 The product of a number and -1 is the opposite of the number.
a•(-1) = -1•a = -a
-2•(-1) = 2
Distributive Property
a(b + c) = ab + ac
3(4 + 2) = 3(4) + 3(2)
Reflexive Property of Equality For any real number a, a = a.
–7y = –7y
This is obvious: anything equals
itself. They used the reflexive property.
Symmetric Property of Equality For any real numbers a and b, if a = b then b = a.
If 10 = y, then y = 10.
When solving an equation, I might rearrange things so I end
up with the variable on the left. But I only switched sides; I
didn't actually change anything: the symmetric property.
Transitive Property of Equality For any real numbers a, b and c, if a = b and b = c, then a = c.
If 3x + 2 = y and y = 8, then 3x + 2 = 8.
You might be torn here between the transitive
property and the substitution property. What they did
here was "cut out the middleman" by removing the "y"
in the middle, so they used the transitive property.
Combining Like Terms!
Suppose you have 4 apples + 2 oranges + 6 apples + 3 oranges
This expression can be simplified by combining the like terms to the following expression:
10 apples + 5 oranges
Like terms in algebra are terms that have exactly the same variables and the variable
is raised to the exact same power.
Examples of Like Terms
2x2 and 6x2
6xy and -4xy
xyz3 and 4xyz3
Examples of NON-Like Terms
2x2 and 6x
6xy and -4xy2
5 and 3x
Example of Combining Like Terms:
1. 4x + 3 y + 3 y2 + 2 x2 + 5y + 7x + 2y2
2. 9y +2x – 3y + 4x + 10z – 12z
WORKING WITH THE DISTRIBUTIVE PROPERTY
Example:
3(2x – 5) + 5(3x +6) =
Since in the order of operations, multiplication comes before addition and subtraction, we must get rid
of the multiplication before you can combine like terms. We do this by using the distributive property:
3(2x – 5) + 5(3x +6)
Original Problem
3(2x) – 3(5) + 5(3x) + 5(6)
Distributive Property
6x – 15 + 15x + 30
Simplify
6x + 15x – 15 + 30
Gather Like Terms
21x + 15
Combine Like Terms
You Try: 5(8x + 3) – 7(4x + 11)
Use the distributive property to rewrite the expression without parentheses.
3 x  4 
1.
2.
2  2 y  1
3.
1
10  15r 
5
Simplify the expression by combining like terms.
12m  5m
4.
5.
6a  2a 2  4a  a 2
6.
10r  6t  2  2t  4r
Simplify the expression completely.
10  b  1  4b
7.
8.
4  4 p  3  4  p  1
9.
1  2  6  3r 
Understanding Check:
1.
2.
Which statement is always true?
A
B
C
D
3.
Which is equivalent to
(5x2 + 4x + 1) + (-7x + 2)?
A
B
C
D
-2x2 + 6x + 1
5x2 – 3x – 1
5x2 – 3x + 3
5x2 + 11x + 3
4.
4+a=4a
a + (-4 + 4) = a + 0
a÷4=4÷a
4–a=a–4
The number -35 is best represented by
which one set of numbers?
A. Real
B. Rational
C. Integer
D. Whole
Algebra 1: Lesson 2.5-2.7 Homework
Types of Real Numbers and the Properties of the Real Number System
Identify the Property Being Illustrated.
1. 8  8  0
2. (3  4)  5  3  (4  5)
3. (5)(0)  0
4. (12.5)(1)  12.5
5. (1.25)(6)  (6)(1.25)
1
6. 3   1
3
7. 4(3  a)  12  4a
8. (4  5)  2  (5  4)  2
9. If y = 4 and 4 = x, then y = x
10. If 3 = y, then y = 3
Simplify the Expression.
11. 6  10x  3
12. 9  2( x  4)
13. 7 x  3(4  2 x)
14. 8 x  3(2 x  1)
15. 7( x  2)  5
16. x(6  2 x)  3x
17. 2(3x  1)  4  2 x
18. 2( x  4)  3
19. ( x  4)  3
20. 5 x(2 x  1)  4 x 2
11
Between What Two Integers Does the Square Root Belong?
21.
24
22.  60
23.
7
24.  98
Put a Check in Each box that Names that Number.
Number
144
 49
5.36
1.35987…
1

3
Real
Rational
Put the Numbers in Order from Least to Greatest.
30.
64 , 5 ,  9 , 2
31.
3 , 5.5 ,  18 , 0
32.
1
1
,  2,  ,
3
3
2
Irrational
Integer
Whole
Chapter 1+2 Quiz Practice
12
Evaluate the Expression.
1.
2 x  3 when x = 6
Perform Order of Operations.
2. 3(5  1)  2
3. 16  (3  1)
2
2
Translate the verbal phrase into an expression, equation or inequality.
4. Three times a number is at least 4.
5. 5 less than twice a number.
Is the given number a solution to the equation or inequality (Yes, No)?
7. r  8  21 , if r = 13
6. 4  m  20 , if m = 4
2
Complete the Chart by Putting a Check in the Box for each that applies to the number.
Number
8.
-16
9.
Real
Rational
13
Put the Numbers in order from least to greatest.
10. 2, 2,1.2, 
1
2
Identify the Property being illustrated.
11. -3 + 3 = 0
12. If 3 = x, then x = 3
Irrational
Integer
Whole
Find the Sum or Difference.
13.
–4 + (-1)
14.
8 – ( -2)
Find the Product or Quotient.
15.
(3)(4)(-2)
16. 28  (7)
Tell Whether the Statement is True or False.
17. If a number is an integer, then the number is also a rational number.
Simplify the Expression.
18. 3(2x + 3) – 5x
19. -6(5x – 3) + 2(2 – 8)
What Two Whole Numbers is the Square Root Between?
20.
68