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Roster form just lists out the elements of a set between two set brackets. For example,
{January, June, July} |
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| To notate that two expressions are equal to each, use the symbol = between them. |
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| a is less than or equal to b |
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| a is greater than or equal to b |
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| A mathematical statement uses the equality and inequality symbols shown above. It can be judged either true or false. |
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| Natural (or Counting) Numbers |
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{0, 1, 2, 3, 4, 5, ...}
The only difference between this set and the one above is that this set not only contains all the natural numbers, but it also contains 0, where as 0 is not an element of the set of natural numbers. |
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Z = {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}
This set adds on the negative counterparts to the already existing whole numbers (which, remember, includes the number 0).
The natural numbers and the whole numbers are both subsets of integers. |
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Q = {rational number| a and b are integers and not equal to}
In other words, a rational number is a number that can be written as one integer over another.
Be very careful. Remember that a whole number can be written as one integer over another integer. The integer in the denominator is 1 in that case. For example, 5 can be written as 5/1.
The natural numbers, whole numbers, and integers are all subsets of rational numbers. |
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I = {x | x is a real number that is not rational}
In other words, an irrational number is a number that can not be written as one integer over another. It is a non-repeating, non-terminating decimal.
One big example of irrational numbers is roots of numbers that are not perfect roots - for example rootor root. 17 is not a perfect square - the answer is a non-terminating, non-repeating decimal, which CANNOT be written as one integer over another. Similarly, 5 is not a perfect cube. It's answer is also a non-terminating, non-repeating decimal.
Another famous irrational number is pi (pi). Even though it is more commonly known as 3.14, that is a rounded value for pi. Actually it is 3.1415927... It would keep going and going and going without any real repetition or pattern. In other words, it would be a non terminating, non repeating decimal, which again, can not be written as a rational number, 1 integer over another integer. |
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R = {x | x corresponds to point on the number line}
Any number that belongs to either the rational numbers or irrational numbers would be considered a real number. That would include natural numbers, whole numbers and integers. |
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Zero, on the number line, is called the origin. It separates the negative numbers (located to the left of 0) from the positive numbers (located to the right of 0).
I feel sorry for 0, it does not belong to either group. It is neither a positive or a negative number. |
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Order Property for
Real Numbers |
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Definition
Given any two real numbers a and b,
if a is to the left of b on the number line, then a < b.
If a is to the right of b on the number line, then a > b. |
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Most people know that when you take the absolute value of ANY number (other than 0) the answer is positive.absolute value
3 is 3 units away from 0
so , |3|= 3
-3 is 3 units away from 0 so, |-3|= 3 |
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| Replace ? with <, >, or = . 3 ? 5 |
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| Since 3 is to the left of 5 on the number line, then 3 < 5. |
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| Replace ? with <, >, or = . 7.41 ? 7.41 |
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| Since 7.41 is the same number as 7.41, then 7.41 = 7.41. |
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| Replace ? with <, >, or = . 2.5 ? 1.5 |
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| Since 2.5 is to the right of 1.5 on the number line, then 2.5 > 1.5. |
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| Is the following mathematical statement true or false? 2 > 7 |
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Since 2 is to the left of 7 on the number line, then 2 < 7.
Therefore, the given statement is false. |
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| Is the following mathematical statement true or false? 5 > 5 |
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| Since 5 is the same number as 5 and the statement includes where the two numbers are equal to each other, then this statement is true. |
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The sentence as a mathematical statement.
2 is less than 5. |
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Definition
Reading it left to right we get:
2 is less than 5
2 < 5 |
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The sentence as a mathematical statement.
10 is less than or equal to 20. |
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Definition
Reading it left to right we get:
10 is less than or equal to 20
10 < 20 |
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The sentence as a mathematical statement.
-2 is greater than -3. |
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Definition
Reading it left to right we get:
-2 is greater than -3
-2 > -3 |
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The sentence as a mathematical statement.
0 is greater than or equal to -1. |
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Definition
Reading it left to right we get:
0 is greater than or equal to -1
0 > -1 |
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The sentence as a mathematical statement.
5 is not equal to 2. |
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Definition
Reading it left to right we get:
5 is not equal to 2
example 10 |
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Term
List the elements of the following sets that are also elements of the given set
{-4, 0, 2.5, pi , example 11b,example 11, 11/2, 7} |
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| Natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. |
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The numbers in the given set that are also natural numbers are
{example 11, 7}.
Note that example 11 simplifies to be 5, which is a natural number. |
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The numbers in the given set that are also whole numbers are
{0, example 11, 7}. |
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| Do you feel good about yourself? |
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| I don't know actually I'm just the guy making these cards |
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Definition
The numbers in the given set that are also integers are
{-4, 0,example 11, 7}. |
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Definition
The numbers in the given set that are also rational numbers are
{-4, 0, 2.5,example 11 , 11/2, 7}. |
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The numbers in the given set that are also irrational numbers are
{pi, example 11b }.
These two numbers CANNOT be written as one integer over another. They are non-repeating, non-terminating decimals. |
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Definition
The numbers in the given set that are also real numbers are
{-4, 0, 2.5, pi, example 11b,example 11, 11/2, 7}. |
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| Replace ? with <, >, or = . |-2.5| ? |2.5| |
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Definition
Since |-2.5| = 2.5 and |2.5| = 2.5, then the two expressions are equal to each other:
|-2.5| = |2.5| |
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| Replace ? with <, >, or = . -3 ? |3| |
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Definition
First of all, |3| = 3.
Since -3 is to the left of 3 on the number line, then -3 < |3|. |
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| Replace ? with <, >, or = . 4 ? |-1| |
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Definition
First of all, |-1| = 1
Since 4 is to the right of 1 on the number line, then 4 > 1 |
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| Since 5 is to the right of 0 on the number line, then 5 > 0. |
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|.5| = |-.5|
They are both are the same distance from zero |
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-2<2
you should know this one |
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| False, 2 is not greater than 4 |
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| 5 is greater than or equal to -5 |
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a/b where b≠0
A numeric fraction is a quotient of two numbers. The top number is called the numerator and the bottom number is referred to as the denominator. The denominator cannot equal 0. |
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A prime number is a whole number that has two distinct factors, 1 and itself.
Examples of prime numbers are 2, 3, 5, 7, 11, and 13. The list can go on and on.
Be careful, 1 is not a prime number because it only has one distinct factor which is 1.
When you rewrite a number using prime factorization, you write that number as a product of prime numbers.
For example, the prime factorization of 12 would be
12 = (2)(6) = (2)(2)(3).
That last product is 12 and is made up of all prime numbers. |
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| When is a Fraction Simplified? |
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| Good question. A fraction is simplified if the numerator and denominator do not have any common factors other than 1. You can divide out common factors by using the Fundamental Principle of Fractions, shown next. |
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| Fundamental Principle of Fractions |
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| In other words, if you divide out the same factor in both the numerator and the denominator, then you will end up with an equivalent expression. An equivalent expression is one that looks different, but has the same value. |
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Writing the Fraction in Lowest Terms
(or Simplifying the Fraction) |
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Definition
p 1: Write the numerator and denominator as a product of prime numbers.
Step 2: Use the Fundamental Principle of Fractions to cancel out the common factors. |
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| write the fraction in lowest terms 7/35 |
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Definition
Write the numerator and denominator as a product of prime numbers.
*Rewrite 35 as a product of primes
Step 2: Use the Fundamental Principle of Fractions to cancel out the common factors.
*Div. the common factor of 7 out of both num. and den.
Note that even though the 7's divide out in the last step, there is still a 1 in the numerator. 7 is thought of as 7 times 1 (not 0). |
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| Write the fraction in lowest terms. 90/50 |
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Definition
Step 1: Write the numerator and denominator as a product of prime numbers.
*Rewrite 90 as a product of primes
*Rewrite 50 as a product of primes
Step 2: Use the Fundamental Principle of Fractions to cancel out the common factors.
*Div. the common factors of 2 and 5 out of both num. and den. |
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