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MTEL Mathematics (Middle School)(47): Practice & Study Guide41 chapters | 321 lessons | 12 flashcard sets

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 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 of two positive numbers is positive.

**Positive + Positive = Positive**

Addition of two negative numbers is negative.

**Negative + Negative = Negative**

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**

If |Negative| > |Positive|

**Positive + Negative = Negative**

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 odd numbers is even.

**Odd + Odd = Even**

Addition of one even and one odd number is odd.

**Even + Odd = Odd**

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

If Positive1 > Positive2

**Positive1 - Positive2 = Positive**

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

If Positive1 < Positive2

**Positive1 - Positive2 = Negative**

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**

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**

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**

Subtracting an odd number from another gives an even number.

**Odd - Odd = Even**

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**

Multiplication of two positive numbers is positive.

**Positive X Positive = Positive**

Multiplication of two negative numbers is positive.

**Negative X Negative = Positive**

Multiplication of one negative and one positive number is negative.

**Negative X Positive = Negative**

and

**Positive X Negative = Negative**

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 two odd numbers is odd.

**Odd X Odd = Odd**

Multiplication of one odd and one even number is even.

**Odd X Even = Even**

and

**Even X Odd = Even**

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

**Positive / Positive = Positive**

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

**Negative / Negative = Positive**

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

**Positive / Negative = Negative**

and

**Negative / Positive = Negative**

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

In this lesson, we saw an overview of the common theorems related to mathematical operations on real numbers.

- The result of the
**addition**of two numbers depends on the sign and magnitude of the numbers. Addition of two even numbers gives an even number. Addition of two odd numbers gives an even number and addition of one even and one odd number gives an odd number. - The result of
**subtraction**also depends on the sign and magnitude of the numbers. Subtracting an even number from another even number results in an even number, subtracting an odd number from another odd number results in an even number and subtracting an even number from an odd number or an odd number from an even number gives an odd number. - The result of
**multiplication**depends only on the sign of the numbers. Multiplication of two even numbers results in an even number, multiplication of two odd numbers results in an odd number, and multiplication of an odd number with an even number results in an even number. - The sign of the result of
**division**also depends on the signs of the numbers and is similar to multiplication.

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Lesson
13 in chapter 12 of the course:

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MTEL Mathematics (Middle School)(47): Practice & Study Guide41 chapters | 321 lessons | 12 flashcard sets

- What are the Different Types of Numbers? 6:56
- What are Real Numbers? - Definition & Properties 4:50
- The Order of Real Numbers: Inequalities 4:36
- Finding the Absolute Value of a Real Number 3:11
- What are Rational Numbers? - Definition & Examples 5:34
- What are Irrational Numbers? - Definition & Examples 6:36
- Comparing & Ordering Rational Numbers 7:07
- Graphing Rational Numbers on a Number Line 5:02
- Adding & Subtracting Rational Numbers 8:12
- Multiplying & Dividing Rational Numbers 6:19
- The Closure Property of Real Numbers
- Inverse Property: Definition & Examples 3:05
- Overview of Common Number Theorems
- Go to MTEL Middle School Mathematics: Real Numbers

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