Download Integers, Rational Numbers, and Properties of

Integers, Rational Numbers,
and Properties of Real
Numbers
Integers
Definition
• Integers are the set of real numbers that consists of counting
numbers, their inverses, and the number zero.
-5 -4 -3 -2 -1 0 1 2 3 4 5
Absolute Values
• Given an integer n, the absolute value of n, represented by |n|,
is the distance between n and 0 on the number line.
• Examples:
o|3| = 3
|3|
-3 -2 -1 0 1 2 3
o|-5| = 5
|-5|
-5 -4 -3 -2 -1 0 1 2
3 4 5
Operations on Integers
• Adding integers:
oPositive + Positive: Add, and you’re done.
Example: 3 + 2 = 5
3
-5
-4
-3
-2
-1
2
0 1 2 3 4 5
oNegative + Negative: Add, then prefix a negative sign to your
sum.
Example: -2 + (-1) = -3
-1
-2
-3
-2
-1
0 1 2 3
Operations on Integers
• Adding integers:
oPositive + Negative: Subtract, then prefix the sign of the number
with the larger absolute value.
Example 1: 3 + (-2) = 1 (Because |3| > |-2|)
-3
-2
-1
-2
3
0 1 2 3
Example 2: -3 + 2 = -1 (Because |-3| > |2|)
2
-3
-3
-2
-1
0 1 2
3
Operations on Integers
• Subtracting Integers: Change the sign of the minuend, and then
proceed with integer addition.
Examples:
o3 – 2 = 3 + (-2) = 1
o3 – (-5) = 3 + 5 = 8
o-5 – 2 = -5 + (-2) = -7
o-5 – (-6) = -5 + 6 = 1
Operations on Integers
• Multiplying and Dividing Integers:
oWhen the two numbers have the same sign, just do the required
operation and you’re done.
6 * 2 = 12
(-6) / (-3) = 2
oWhen the two numbers have opposite signs, do the required
operation, then prefix a negative sign to the result.
5 * (-4) = -20
(-30) / 6 = -5
Operations on Integers
• Raising an integer to an exponent:
oPositive Integers: Just do the operation and you’re done.
23 = 8
52 = 25
oNegative Integers: If the exponent is even, just do the operation.
If the exponent is odd, do the operation, then prefix a negative
sign to the result.
(-3)2 = 9
(-5)3 = -125
Word Problems on Integers
• Height Problems
• Age Problems
• Consecutive Integer Problems
• Digit Problems
Problem 1:
• The highest elevation in North America is Mt. McKinley, which is
20,320 feet above sea level. The lowest elevation is Death
Valley, which is 282 feet below sea level. What is the distance
from the top of Mt. McKinley to the bottom of Death Valley?
(Taken From:
http://www.mathgoodies.com/lessons/vol5/intro_integers.html)
Problem 1 Solution:
• We know that:
Height of Mt. McKinley = 20,320 feet above sea level = 20,320
feet
Height of Death Valley = 282 feet below sea level = -282 feet
• To get the distance from the bottom of Death Valley to the top of
Mt. McKinley, we just have to add their absolute values
together:
|20,320ft| + |-282ft| = 20,320ft + 282ft = 20,602ft
Problem 2:
• In January of the year 2000, I was one more than eleven times
as old as my son William. In January of 2009, I was seven more
than three times as old as him. How old was my son in January
of 2000?
(Taken From: http://purplemath.com/modules/ageprobs.htm)
Problem 2 Solution:
• In January of the year 2000, I was one more than eleven times
as old as my son William. In January of 2009, I was seven more
than three times as old as him. How old was my son in January
of 2000?
• We are faced with this given:
Years elapsed = 2009 – 2000 = 9
January 2000
January 2009
William
x
x+9
Father
11x + 1
(11x + 1) + 9 = 11x + 10
Problem 2 Solution:
• In January of the year 2000, I was one more than eleven times
as old as my son William. In January of 2009, I was seven more
than three times as old as him. How old was my son in January
of 2000?
• Also, in 2009, the father’s age is seven more than three times
as old as William.
• 11x + 10 = 3(x + 9) + 7
January 2000
January 2009
William
x
x+9
Father
11x + 1
(11x + 1) + 9 = 11x + 10
Problem 2 Solution:
• 11x + 10 = 3(x + 9) + 7
• 11x + 10 = 3x + 27 + 7
• 11x + 10 = 3x + 34
• 11x – 3x = 34 – 10
• 8x = 24
• x = 3 (William’s age in January 2000)
• Final answer: William was 3 years old in January 2000.
Problem 3:
• The sum of three consecutive integers is 36. What are those
integers?
Problem 3 Solution:
• We can write this problem as:
• x + (x + 1) + (x + 2) = 36
• 3x + 3 = 36
• 3x = 36 – 3
• 3x = 33
• x = 11
• x + 1 = 12
• x + 2 = 13
Problem 4:
• The sum of the digits of a two-digit number is 7. When the digits
are reversed, the number is increased by 27. Find the number.
(Taken from: http://www.purplemath.com/modules/systprob.htm)
Problem 4 Solution:
• The sum of the digits of a two-digit number is 7. When the digits
are reversed, the number is increased by 27. Find the number.
• First Condition: The sum of the digits of a two-digit number is 7.
• Equation 1: x + y = 7  x = 7 - y
• Second Condition: When the digits are reversed, the number is
increased by 27
• Equation 2: 10y + x = (10x + y) + 27  10y – y + x – 10x = 27
 9y – 9x = 27  9(y – x) = 27  y – x = 3
Problem 4 Solution:
• The sum of the digits of a two-digit number is 7. When the digits
are reversed, the number is increased by 27. Find the number.
• y – (7 – y) = 3
•y–7+y=3
•y+y=3+7
• 2y = 10
•y=5
•x=7–y=7–5=2
• Answer: The number is 25.
Rational Numbers
Rational Numbers
• A number can be called rational if it can be written as a quotient
of two integers.
• Examples:
2=2/1
0.5 = 1 / 2
0.3333… = 1 / 3
Irrational Numbers
• Any number that does not satisfy the said condition is called
irrational.
• Two Types:
oIrrational Constants (Examples: Pi and e)
oRadicals that cannot be written as a whole number (Examples:
√2 and 3√10)
Properties of Irrational Numbers
• n√xy = n√x * n√y
• n√(x / y) = n√x / n√y
• (n√x)n = x
• n√(m√x) = nm√x
• an√x (+ or -) bn√x = [a (+ or -) b] n√x
Simplifying Irrational Numbers
• Simplify √12.
Simplifying Irrational Numbers
• Simplify √12.
• √12 = √(4 * 3) = √4 * √3 = 2√3
Rationalizing Irrational Numbers
• Simplify 3 / √7.
Rationalizing Irrational Numbers
• Simplify 3 / √7.
• (3 / √7) * (√7 / √7)
• 3√7 / 7
Rationalizing Irrational Numbers
• Simplify (√2 + √8) / (√5 – 8).
Rationalizing Irrational Numbers
• Simplify (√2 + √8) / (√5 – 8).
• (√2 + √(4 * 2)) / (√5 – 8)
• (√2 + √4 * √2) / (√5 – 8)
• (√2 + 2√2) / (√5 – 8)
• (3√2) / (√5 – 8)
• [(3√2) * (√5 + 8)] / [(√5 – 8) * (√5 + 8)]
• (3√(2 * 5) + (3√2 * 8)) / (5 – 64)
• (3√10 + 24√2) / (-59)
Properties of Real
Numbers
Properties of Real Numbers
(All info after this slide are courtesy of:
http://www.math.com/school/subject2/lessons/S2U2L1DP.html)
Properties of Real Numbers
1) Commutative Property
Addition:
5a + 4 = 4 + 5a
Multiplication:
3 x 8 x 5b = 5b x 3 x 8
Properties of Real Numbers
2) Associative Property
Addition:
(4x + 2x) + 7x = 4x + (2x + 7x)
Multiplication:
2x2(3y) = 3y(2x2)
Properties of Real Numbers
3) Distributive Property
2x(5 + y) = 10x + 2xy
4) Density property
- It simply states that given two real numbers, we can find
infinitely other real numbers that lie between them.
Properties of Real Numbers
5) Identity property
Addition:
5y + 0 = 5y
Multiplication:
2c × 1 = 2c
Imaginary Numbers
• An imaginary number is any number that is not a real number.
• Famous example: i = √-1
Imaginary Numbers
• We notice a pattern when we raise i to an exponent n:
• i1 = i
• i2 = -1
• i3 = -i
• i4 = 1
• And so on…
Imaginary Numbers
• In general, the value of the imaginary number i raised to an
exponent n, where n is an integer and n >= 0, is given by:
• in = i(n mod 4)
• Note: n mod 4 means that you should take the remainder when
n is divided by 4.
Problem 1:
• Evaluate i8.
Problem 1 Solution:
• Evaluate i8.
• 8 mod 4 = 0
• Thus, i0 = i4 = i8 = 1.
Problem 2:
• Evaluate: (i + 2)2.
Problem 2 Solution:
• Evaluate: (i + 2)2.
• (i + 2)2 = i2 + 2i + 4
• (-1) + 2i + 4
• 2i + 3
Sources: