Addition and subtraction of directed numbers - Examples, Exercises and Solutions

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
    64=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.
    +64=+2+6-4=+2
    6+4=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
    6(4)=+106-(-4)=+10

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

Suggested Topics to Practice in Advance

  1. Positive and negative numbers and zero
  2. Opposite numbers
  3. Elimination of Parentheses in Real Numbers
  4. Real line or Numerical line

Practice Addition and subtraction of directed numbers

Exercise #1

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 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 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

The sum of two numbers is positive.

Therefore, the two numbers are...?

Step-by-Step Solution

Testing through attempts:

 

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

1+2 = 3

Positive result.

 

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

-1+(-2) = -3

Negative result.

 

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.

 

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 #3

a is a positive number.

b is a negative number.

The sum of a+b is...?

Step-by-Step Solution

We will test this through experiments:

Let's assume that the value of the positive number is greater than the value of the negative number 1 and 2.

1+(-2) = -1

The result is negative.

We will try to make the value of the second number greater than the first 2 and 1.

2+(-1)= 1

The result is positive.

That is, we can see that the result depends on the values of the two numbers, so we cannot know from the beginning what the result will be.

Answer

It is not possible to know.

Exercise #4

a is a positive number.

b is a negative number.

What kind of number is the sum of b and a?

Step-by-Step Solution

We will illustrate with an example:

Let's assume that a is 1 and b is -2

 

1+ (-2) = 
1-2 = -1

Answer: Negative

 

Now we define that a is 2

and b is -1

2+(-1) = 
2-1 = 1

Even though the operation is negative, the number remains positive.

That is, if the absolute value of the positive number (a) is greater than that of the negative (b), the result will still be positive.

As we do not have data on this information, it is impossible to know what the sum of a+b will be.

Answer

.Impossible to know.

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

 

Now we replace in the exercise:

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

In this case, the result is positive!

 

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

We define that

a = -2

b = -1

 

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

 

In this case, the result is negative!

 

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

Answer

Impossible to know.

Exercise #1

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.

 

Now let's place them in an exercise:

 

2-1=1

The result is positive!

 

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!

 

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 #2

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!

 

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

We define that

a = -2

b = 1

 

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

 

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 #3

27(7)+(6)+211= -27-(-7)+(-6)+2-11=

Video Solution

Step-by-Step Solution

First, we solve the multiplication exercise, that is where there is a plus or minus sign before another sign.

27+76+211= -27+7-6+2-11=

Now we solve as a common exercise from left to right:

27+7=20 -27+7=-20

206=26 -20-6=-26

26+2=24 -26+2=-24

2411=35 -24-11=-35

Answer

35 -35

Exercise #4

(2)+3= (-2)+3=

-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333444555666

Video Solution

Answer

1 1

Exercise #5

5+(2)= 5+(-2)=

-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333444555666

Video Solution

Answer

3 3

Exercise #1

5(2)= -5-(-2)=

-7-7-7-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333444555666777

Video Solution

Answer

3 -3

Exercise #2

4+(2)= -4+(-2)=

-7-7-7-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333444555777666

Video Solution

Answer

6 -6

Exercise #3

3+(4)= 3+(-4)=

-7-7-7-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333444555777666

Video Solution

Answer

1 -1

Exercise #4

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

-7-7-7-6-6-6-5-5-5-4-4-4-3-3-3-2-2-2-1-1-1000111222333444555777666

Video Solution

Answer

5 5

Exercise #5

14(3)= 14-(-3)=

Video Solution

Answer

17 17

Topics learned in later sections

  1. Multiplication and Division of Real Numbers
  2. Integers