# Comparing and Ordering Rational Numbers: TEAS

## Basic Terms and Terminology Relating to Comparing and Ordering Rational Numbers

- Positive number: ALL numbers greater than zero in addition to zero.
- Negative number: ALL numbers less than zero

## Positive Numbers

Positive numbers are ALL numbers greater than zero in addition to zero. We typically use mostly positive numbers in our daily lives and to solve most everyday problems. All of the basic arithmetic calculations above, including addition, subtraction multiplication and division were done using positive numbers.

The numbers on the number line above towards the right show positive numbers from zero to nine. Positive numbers are written as numbers without any special designation to indicate that they are positive numbers.

**For example, positive numbers are simply written as:**

- 16
- 357
- 8,908

And NOT as

- + 6
- + 357
- + 8908

## Negative Numbers

Negative numbers, on the other hand, are ALL numbers less than zero, as shown on the number line above towards the left. Negative numbers are far less encountered in our daily lives than positive numbers.

When the thermometer registers as 10 degrees below zero the temperature is - 10 degrees and when you overdraw on your checking account, you balance may be - $67.08 which means you owe the bank $67.08 before your bank account can have a $0 balance and it will take more than $67.08 to get your checking account balance above zero.

Negative numbers have the special designation of minus (-) to indicate that the number is a negative number less than zero and NOT a positive number.

Additionally, when you subtract a larger positive number from a smaller positive number, the answer to the calculation is a negative number, as you can see in the examples below.

2 minus 4 or

2

__- 4
__- 2

4 minus 12 or

4

__- 12
__ - 8

**Examples of negative numbers are:**

- -9
- -34
- -675
- -7,980

Mathematical calculations with negative numbers are different from those mathematical calculations done with positive numbers because the meanings of negative numbers as being less than zero are very different from the meanings of positive numbers from zero upward to infinity.

## Addition with Positive and Negative Numbers

The same concepts about addition that were previously discussed with basic addition still hold true when you are adding negative numbers together or adding one or more positive numbers with one or more negative numbers.

The mechanics of addition calculations with negative numbers, however, is different simply because negative numbers are not conceptually the same as positive numbers. Positive and negative numbers are different, therefore, these calculations are different.

As you know, when you add positive numbers together, the answer will be a positive number. For example, 4 + 5 = 9 and 11 + 5 = 16, as shown in the diagram above. Notice that there are no special designations for the positive numbers of 9 and 16 that were the sums for these simple addition examples.

When you add a negative number to one or more negative numbers, the answer to the calculation is a negative number, as shown in the diagram above. For example, the addition of all of these negative numbers will add up to a negative number:

- 9
- 4
- 3
__5__- 21 or negative 21

When you add a negative number to a positive number the sum, or the answer to the calculation, can be either a positive or a negative number, as shown in the diagram above.

The sum of negative and positive numbers is a positive number when the sum of the positive numbers is greater than the sum of the negative numbers, as shown in the diagram above. Here are some examples:

Add 6 + 3 - 2 - 6 + 4 + 5

In the above example, all of the positive numbers (5, 3 and 4) add up to 12 and all the negative numbers (- 2 and - 6) add up to -8. Because the sum of the positive numbers is greater than the sum of the negative numbers, the sum of all of these numbers will be a positive number and the answer is positive 5 because the sum of the positive numbers of 12 is 5 more than the sum of the negative numbers which is -8.

On the other hand, the sum of negative and positive numbers is a negative number when the sum of the negative numbers is greater than the sum of the positive numbers, as shown in the picture above. Here is an example:

Add 6 - 3 - 2 - 6 + 4 = -1

In the above example, all of the positive numbers (6 and 4) add up to 10 and all the negative numbers (- 3, - 2 and - 6) add up to - 11. Because the sum of the negative numbers is greater than the sum of the positive numbers, the sum of all of these numbers will be a negative number and the answer is negative 5 because the sum of the negative numbers of 11 is 1 more than the sum of the positive numbers which is - 10.

The sum of negative and positive numbers will be a zero when the sum of the positive numbers is equal to the sum of the negative numbers. Here is an example:

Add 6 - 3 -2 -5 + 4 = 0

In the above example, all of the positive numbers (6 and 4) add up to 10 and all the negative numbers (- 3, - 2 and - 5) add up to - 10. Because the sum of the negative numbers is the same as and identical to the sum of the positive numbers, the sum of all of these numbers will be a zero, which is considered a positive number.

## Subtraction with Positive and Negative Numbers

Again, the same concepts about subtraction that were previously discussed with basic subtraction still hold true when you are subtracting with negative numbers. The mechanics of the subtraction of negative numbers, however, is different simply because negative numbers are not conceptually the same as positive numbers. Positive and negative numbers are different, therefore, these calculations are different.

**Here are some examples of subtraction with positive and negative numbers:**

Subtracting a positive number from a positive number is the same as the subtraction that you learned above when the first number is larger or greater than the second number. Here is an example:

+ 6 minus + 4 or 6 - 4 = 2

Subtracting a positive number from a positive number is different from regular subtraction when the first number is smaller or less than the second number. The answer to these kinds of calculations will be a negative number. Here is an example:

+ 6 minus 10 or 6 - 10 = - 4 or negative 4

Subtracting a positive number from a positive number is the same as regular subtraction when the first number is equal to the second number. The answer to these kinds of calculations will always be zero. Here is an example:

10 minus - 10 = 0

Subtracting a negative number from a negative number will give a negative number as the answer or difference. Simply stated using the checkbook story above, you checking account will be further withdrawn when you write a check for $40 and your balance before this check was - $25. After this check is written your new balance is - $65 and you are further in debt.

Here are some examples:

- (- 7) - (- 4) = -11
- (- 17) - (- 14) = -31
- (- 700) - (- 401) = - 1,101

Did you notice a pattern with the answers above? Did you notice that all these subtraction calculations with 2 negative numbers not only has a negative number as the answer, but the answer is really the sum of the 2 negative numbers in the calculation question.

For example, since 7 plus 4 is 11, (- 7) - (- 4) = -11 and

since 17 plus 14 is 31, (- 17) - (- 14) = -31

since 700 plus 401 is 1,101, (- 700) - (- 401) = - 1,101

## Multiplication with Positive and Negative Numbers

**The rules relating to the multiplication of positive and negative numbers are:**

- The answer or product to the multiplication calculation is a positive number when the two multipliers are both positive and when the two multipliers are positive.
- The answer or product to the multiplication calculation is a positive number when the two multipliers are both negative and when the two multipliers are both negative.
- The answer or product to the multiplication calculation is a negative number when one multiplier is positive and the other is negative.

Examples when the multipliers are all positive:

- (+ 8) x (+7) = + 56 or 56
- (+ 12) x (+4) = + 48 or 48
- (+ 8) x (+2) x (+3) = + 48 or 48

Examples when the multipliers are all negative:

- (- 8) x (-7) = + 56 or 56
- (- 12) x (-4) = + 48 or 48
- (- 8) x (-2) x (-3) = + 48 or 48

Examples when at least one multiplier is positive and at least one multiplier is negative:

- (- 8) x (+7) = - 56
- (12) x (-4) = - 48
- (8) x (-2) x (-3) = - 48

## Division with Positive and Negative Numbers

**The rules relating to the division of positive and negative numbers, which is similar to the multiplication of positive and negative numbers, are:**

- The answer or quotient to the division calculation is a positive number when both the divisor and the dividend are positive.
- The answer or quotient to the division calculation is a positive number when both the divisor and the dividend are negative.
- The answer or quotient to the division calculation is a negative number when the divisor is negative and the dividend is positive AND when the divisor is positive and the dividend is negative.

Simply stated, when the divisor and dividend are the same in terms of their sign (+ or -), the answer to the division calculation is always positive; and when the divisor and dividend are different in terms of their sign (+ or -), the answer to the division calculation is always negative.

**Examples of division when the divisor and dividend are both positive:**

- (+12) / (+4) = + 48 or 48
- (+6) / (+3) = + 2 or 2
- (+99) / (+3) = + 33 or 33
- (+6.33) / (+3) = + 2.11 or 2.11

Examples of division when the divisor and dividend are both negative:

- (-12) / (-4) = + 48 or 48
- (-6) / (-3) = + 2 or 2
- (-99) / (-3) = + 33 or 33
- (-6.33) / (-3) = + 2.11 or 2.11

Examples of division when the divisor is negative and the dividend is positive AND when the divisor is positive and the dividend is negative.

- (-12) / (4) = - 3
- (-6) / (3) = - 2
- (12) / (-4) = - 3
- (99) / (-3) = - 33
- (-6.33) / (-3) = - 2.11

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