Addition and Subtraction of Real Numbers

πŸ†Practice addition and subtraction of directed numbers

The addition and subtraction of real numbers are based on certain key principles. All principles will be explained using two real numbers, but certainly, the numbers in the exercise do not influence the method of resolution, therefore, these principles can be applied to any number in the exercise.

A1 - Addition and Subtraction of Real Numbers

  • When we have two real numbers with the same sign (plus or minus), this sign will remain in the result, which will, in fact, be the result of the addition. That is, if both numbers have a plus sign the result of the addition will also be positive. If both numbers have a minus sign the result of the subtraction will also be negative.
    +6+4=+10+6+4=+10
    βˆ’6βˆ’4=βˆ’10-6-4=-10

  • When we have two numbers with different signs it is crucial to determine which of the two has the greater absolute value (absolute: the distance from zero). The larger number will determine the sign of the result and, in fact, we will perform a subtraction operation.
    +6βˆ’4=+2+6-4=+2
    βˆ’6+4=βˆ’2-6+4=-2

  • When we have an exercise with a sequence of two signs (usually separated by parentheses) we will differentiate between several cases:

  • When the sequence is of two plus signs the result will also be positive
    6+(+4)=+106+(+4)=+10

  • When the sequence is of two minus signs the result will also be positive
    6βˆ’(βˆ’4)=+106-(-4)=+10

  • When the sequence is of minus and plus or of plus and minus the result will be negative.
    6+(βˆ’4)=+26+(-4)=+2
    6βˆ’(+4)=+26-(+4)=+2

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Test yourself on addition and subtraction of directed numbers!

einstein

\( (+6)-(+11)= \)

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Association of Operations in Exercises with Addition and Subtraction of Real Numbers

After studying real numbers, it's time to learn how to use them in an equation. Initially, our goal with equations is to simplify them to make it easier to solve problems, and we do this by grouping operations and adding and subtracting real numbers. We just need to remember two rules:

  • When the mathematical operation and the sign of the following real number are of the same type, we group them into a sum.
  • For example: 5+(+5)5+(+5) / 5βˆ’(βˆ’5)5-(-5)
    5+55+5 will become
  • When the mathematical operation and the sign of the following real number are of different types, we group them into an operation that will give the difference between them. For example: 5+(βˆ’5)5+(-5) / 5βˆ’(+5)5-(+5)
    5βˆ’55-5 will become

For example:

10+(+5)βˆ’(+3)βˆ’(βˆ’6)+(βˆ’8)=10+(+5)-(+3)-(-6)+(-8)=
10+5βˆ’3+6βˆ’8=1010+5-3+6-8=10


The Elevator Method for Adding and Subtracting Real Numbers

There is a very well-known tactic that helps to understand the topic of real numbers in the best way, it's called the elevator method and it serves to clarify the addition and subtraction of real numbers. With this method, we imagine that the exercise is like a journey in an elevator that goes through the floors. Observe the following exercise:

βˆ’5βˆ’(+1)βˆ’(βˆ’8)+(βˆ’3)=-5-(+1)-(-8)+(-3)=

Before using the elevator method we have to group the signs to simplify the exercise

βˆ’5βˆ’1+8βˆ’3=-5-1+8-3=

Now look at the first number. In fact, you start the exercise on floor -5 and now you are asked to go down one floor. This way you reach floor -6.
Now you are asked to go up 8 floors. So, if we were on floor -6 we will arrive at floor 2. Finally, you are asked to go down 3 floors, therefore, you end up on floor -1, which is the result of the exercise

βˆ’5βˆ’1+8βˆ’3=βˆ’1-5-1+8-3=-1

We will give consistency to the principles outlined through the following examples:

(+3)+(+4)+(+5)=3+4+5=+12(+3) + (+4) + (+5) = 3+4+5= +12

(βˆ’3)+(βˆ’4)+(βˆ’5)=βˆ’3βˆ’4βˆ’5=βˆ’12(-3) + (-4) + (-5) = -3-4-5= -12

βˆ’10+2=βˆ’8-10+2= -8

6βˆ’20=βˆ’146-20= -14

(βˆ’10)βˆ’(βˆ’100)=βˆ’10+100=90(-10)-(-100)= -10+100= 90

8+(βˆ’4)=8βˆ’4=48+(-4)= 8-4= 4


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Exercises on Addition and Subtraction of Real Numbers

Exercise 1

Assignment

βˆ’27βˆ’(βˆ’7)+(βˆ’6)+2βˆ’11= -27-\left(-7\right)+\left(-6\right)+2-11=

Solution

First, we resolve the points that have a plus or minus sign before another sign.

βˆ’27+7βˆ’6+2βˆ’11= -27+7-6+2-11=

Now we solve it as a common exercise:

βˆ’27+7βˆ’6+2βˆ’11=βˆ’35 -27+7-6+2-11=-35

Answer

βˆ’35-35


Exercise 2

Assignment

?βˆ’(βˆ’12)=βˆ’40 \text{?}-(-12)=-40

Solution

First, let's note that the two minuses turn into a plus.

?+12=βˆ’40 \text{?+}12=-40

We will move the 12 12 to the right side

?=βˆ’40βˆ’12 \text{?}=-40-12

Finally, we solve

?=βˆ’52 \text{?}=-52

Answer:

βˆ’52 -52


Do you know what the answer is?

Exercise 3

Assignment

βˆ’36+6= -36+6=

Solution

We use the laws of addition and subtraction to solve accordingly.

βˆ’36+6=βˆ’30 -36+6=-30

Answer:

βˆ’30 -30


Exercise 4

12βˆ’(βˆ’2)= 12-\left(-2\right)=

Solution

Pay attention to the fact that the minus and minus signs become plus, and we solve the exercise accordingly.

12+2=14 12+2=14

Answer

14 14


Check your understanding

Exercise 5

Assignment

Given:

a a Negative number

b b Negative number

What is the sum of a+b a+b ?

Solution

When we add two negative numbers, the result we will obtain is a negative number.

Answer

Negative


Do you think you will be able to solve it?

examples with solutions for addition and subtraction of directed numbers

Exercise #1

βˆ’3+(?)=5 -3+(?)=5 -5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333444555

Video Solution

Step-by-Step Solution

To find out how much we need to add to the number negative 3 in order to get 5, we will count the number of steps between the two numbers.

Also, we will pay attention to which direction we moved, if we moved to the right then the number is positive, if we moved to the left the number will be negative.

We will start from the number negative 3, and move to the right until we reach the number 5, with each step representing one whole number, as follows:

-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333444555

We discover that the number of steps is 8. Since we moved to the right, the number is positive

Answer

8 8

Exercise #2

A and B are positive numbers.

Therefore, A - B results in...?

Step-by-Step Solution

Let's define the two numbers as 1 and 2.

Β 

Now let's place them in an exercise:

Β 

2-1=1

The result is positive!

Β 

Now let's define the numbers in reverse as 2 and 1.

Let's place an equal exercise and see:

1-2=-1

The result is negative!

Β 

We can see that the solution of the exercise depends on the absolute value of the numbers, and which one is greater than the other,

Even if both numbers are positive, the subtraction operation between them can lead to a negative result.

Answer

Impossible to know

Exercise #3

The sum of two numbers is positive.

Therefore, the two numbers are...?

Step-by-Step Solution

Testing through attempts:

Β 

Let's assume both numbers are positive: 1 and 2.

1+2 = 3

Positive result.

Β 

Let's assume both numbers are negative -1 and -2

-1+(-2) = -3

Negative result.

Β 

Let's assume one number is positive and the other negative: 1 and -2.

1+(-2) = -1

Negative result.

Let's test a situation where the value of the first number is greater than the second: -1 and 2.

2+(-1) = 1

Positive result.

Β 

That is, we can see that when both numbers are positive, or in certain types of cases when one number is positive and the other negative, the sum is positive.

Answer

Answers a+c are correct.

Exercise #4

a is a negative number.

b is a positive number.

Therefore, a - b is....?

Step-by-Step Solution

We test using an example:

We define that

a = -1

b = 2

Now we replace in the exercise:

-1-(2) = -1-2 = -3

In this case, the result is negative!

Β 

We test a case where the value of b is less than a

We define that

a = -2

b = 1

Β 

-2-(1) = -2-1 = -3

Β 

In this case, the result is again negative.

Since it is not possible to produce a case where a is greater than b (because a negative number is always less than a positive number),

The result will always be the same: "negative", and that's the solution!

Answer

Negative

Exercise #5

a and b are negative numbers.

Therefore, what kind of number is is a-b?

Step-by-Step Solution

We test using an example:

We define that

a = -1

b = -2

Β 

Now we replace in the exercise:

-1-(-2) = -1+2 = 1

In this case, the result is positive!

Β 

We test the opposite case, where b is greater than a

We define that

a = -2

b = -1

Β 

-2-(-1) = -2+1 = -1

Β 

In this case, the result is negative!

Β 

Therefore, the correct solution to the whole question is: "It's impossible to know".

Answer

Impossible to know.

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