The Numbers 0 and 1 in Operations

🏆Practice special cases (0 and 1, inverse, fraction line)

The numbers 0 0 and 1 1 have some special characteristics when performing basic operations like addition, subtraction, multiplication, and division—including combined calculations.

In this article we will learn what they are and why they are important.

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Test yourself on special cases (0 and 1, inverse, fraction line)!

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\( 12+3\times0= \)

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Characteristics of 0

When we add zero to a number, it will remain unchanged because no value has been added to it.
5+0=5 5+0=5


The same happens when we subtract 0 0 from a number: the number does not change because we are not taking anything away from it.
 50=5\ 5-0=5


In multiplications, the result will always be 0 0 .
 50=0\ 5 \cdot 0=0


We can summarize multiplication by 0 0 as follows (whether it is multiplied by a positive or negative number):


 0a=0\ 0\cdot a=0 and  a0=0\ a\cdot 0=0.

Even when dividing 0 0 by another number, the result will always be 0 0 .
 0:5=0\ 0:5=0

 0:12=0\ 0:\frac{1}{2}=0

 0:1000=0\ 0:1000=0


 0a=0\ {0 \over a}=0 and  0:a=0\ 0: a=0 (Assuming that (a) is not 0 0 ).


Characteristics of 1

For addition and subtraction operations, 1 adds or subtracts one unit to the number.

 5+1=6\ 5+1=6, 51=4\ 5-1=4

 1+1=2\ 1+1=2, 11=0\ 1-1=0

 10+1=11\ 10+1=11, 101=9\ 10-1=9


For multiplications, when a number is multiplied by  1\ 1 it will not change.

 51=5\ 5 \cdot 1=5

 2531=253\ 253 \cdot 1=253

 10.0001=10.000\ 10.000 \cdot 1=10.000


Something very similar happens with division: if we divide a number by one, the number remains unchanged.

 5:1=5\ 5:1=5

 200:1=200\ 200:1=200

 1.000.000:1=1.000.000\ 1.000.000:1=1.000.000


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Example with Combined Operations

After reviewing the characteristics of 0 and 1, let's try to use them to solve combined exercises according to the order of operations.

Example 1

Calculate (10:2+36)(322×4) \left(10:2+3-6\right)\left(3^2-2×4\right) .

Solution:

We perform the division inside the first parenthesis before the power inside the second parenthesis.

(5+36)(92×4) \left(5+3-6\right)\left(9-2×4\right)

We perform the additions and subtractions inside the first parenthesis (from left to right) and then the multiplication inside the second parenthesis.

(86)(98) \left(8-6\right)\left(9-8\right)

We perform the subtractions.

(2)(1) \left(2\right)\left(1\right)

We have obtained a multiplication of a number by one. Therefore, the result is:

2 2


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Review Questions

What are 1 and 0?

They are numbers that have very important characteristics when we perform operations with them. 1 is known as multiplicative neutral and zero is known as additive neutral. This is because the numbers remain the same when you multiply them by one or add zero to them.


What does 0 mean in mathematics?

Zero in mathematics is used to represent the null value or absence.


Check your understanding

What kind of number is 0?

Zero is an integer. It is neither positive nor negative and is used to represent the null value or as an origin in various situations.


How much is a number divided by 0?

Division by zero is not defined.


Do you think you will be able to solve it?

Examples with solutions for The Numbers 0 and 1 in Operations

Exercise #1

Solve the following exercise:

12+30= 12+3\cdot0=

Step-by-Step Solution

According to the order of operations, we first multiply and then add:

12+(30)= 12+(3\cdot0)=

3×0=0 3\times0=0

12+0=12 12+0=12

Answer

12 12

Exercise #2

Solve the following exercise:

2+0:3= 2+0:3=

Step-by-Step Solution

According to the order of operations rules, we first divide and then add:

2+(0:3)= 2+(0:3)=

0:3=0 0:3=0

2+0=2 2+0=2

Answer

2 2

Exercise #3

25+2510= \frac{25+25}{10}=

Video Solution

Step-by-Step Solution

Let's begin by multiplying the numerator:

25+25=50 25+25=50

We obtain the following fraction:

5010 \frac{50}{10}

Finally let's reduce the numerator and denominator by 10 and we are left with the following result:

51=5 \frac{5}{1}=5

Answer

5 5

Exercise #4

0:7+1= 0:7+1=

Video Solution

Step-by-Step Solution

According to the order of operations rules, we first divide and then add:

0:7=0 0:7=0

0+1=1 0+1=1

Answer

1 1

Exercise #5

12+1+0= 12+1+0=

Video Solution

Step-by-Step Solution

According to the order of operations rules, since the exercise only involves addition operations, we will solve the problem from left to right:

12+1=13 12+1=13

13+0=13 13+0=13

Answer

13

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