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Overview of Common Number Theorems

Instructor: Usha Bhakuni

Usha has taught high school level Math and has master's degree in Finance

In this lesson, we will learn about and look at examples of the common theorems related to the basic mathematical operations of addition, subtraction, multiplication, and division.

Real Numbers

Real numbers include almost any number that you can think of - whole numbers (0, 1, 2, 3), positive numbers (4, 5), negative numbers (-9, -10), rational numbers (3/5, 1.28), and irrational numbers (22/7).

In this lesson, we will learn some basic theorems that apply to the mathematical operations of addition, subtraction, multiplication, and division over the set of real numbers.

Let's play with some numbers!

Addition

Addition of two positive numbers is positive.

Positive + Positive = Positive


Addition


Addition of two negative numbers is negative.

Negative + Negative = Negative


Addition of two negative numbers


Addition of one negative and one positive number is positive if the absolute value of the positive number is larger than the absolute value of the negative number. It is negative if the opposite is true.

If |Positive| > | Negative|

Positive + Negative = Positive


Addition

Addition


If |Negative| > |Positive|

Positive + Negative = Negative


Addition

Addition


Therefore, we can see that the result of the addition of two numbers with different signs depends on both the magnitude (absolute value) and the sign of the numbers.

Addition of two even numbers is even.

Even + Even = Even


Addition of two even numbers


Addition of two odd numbers is even.

Odd + Odd = Even


Addition of two odd numbers


Addition of one even and one odd number is odd.

Even + Odd = Odd


Addition of one even and one odd number

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Subtraction

Subtracting a smaller positive number from another positive number results in a positive number.

If Positive1 > Positive2

Positive1 - Positive2 = Positive


Subtraction


Subtracting a larger positive number from a smaller positive number results in a negative number.

If Positive1 < Positive2

Positive1 - Positive2 = Negative


Subtraction


Now we need to look at the absolute value of the numbers involved.

Subtracting a negative number with a smaller absolute value from a negative number with a larger absolute value results in a negative number.

If |Negative1| > |Negative2|

Negative1 - Negative2 = Negative


Subtraction


Subtracting a negative number with a larger absolute value from a negative number with a smaller absolute value results in a positive number.

If |Negative1| < |Negative2|

Negative1 - Negative2 = Positive


Subtraction


Therefore, we can see that the result of subtraction of two numbers with different signs depends on both the magnitude (absolute value) and the sign of the numbers.

Subtracting an even number from another gives an even number.

Even - Even = Even


Subtraction


Subtracting an odd number from another gives an even number.

Odd - Odd = Even


Subtraction


Subtracting an even number from an odd number or an odd number from an even number gives an odd number.

Even - Odd = Odd

and

Odd - Even = Odd


Subtraction

Subtraction


Multiplication

Multiplication of two positive numbers is positive.

Positive X Positive = Positive


Multiplication


Multiplication of two negative numbers is positive.

Negative X Negative = Positive


Multiplication


Multiplication of one negative and one positive number is negative.

Negative X Positive = Negative

and

Positive X Negative = Negative


Multiplication

Multiplication


Therefore, it can be seen that the sign of the result of multiplication depends only on the signs of the two numbers.

Multiplication of two even numbers is even.

Even X Even = Even


Multiplication of even numbers


Multiplication of two odd numbers is odd.

Odd X Odd = Odd


multiplication of odd numbers


Multiplication of one odd and one even number is even.

Odd X Even = Even

and

Even X Odd = Even


Multiplication

Multiplication


Division

Dividing one positive number by another positive number gives a positive number.

Positive / Positive = Positive


Division


Dividing one negative number by another negative number gives a positive number.

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