Negative Numbers: Definition, Examples, Properties, Operations

Negative Numbers: In daily life, we deal with lots of numbers. A number is a mathematical concept used to express a quantity and count or calculate. We can classify the numbers into different types: whole numbers, natural numbers, integers, real numbers, rational, irrational numbers, etc.

To make counting easier, humans invented natural numbers. After that, they invented zero that provides us with the concept of whole numbers and then negative numbers, real numbers, etc. Negative numbers help us show the temperature, which is less than zero, to show the overdrawn bank balance, etc. Let us see the application and properties of negative numbers in this article.

What are Numbers?

Numbers are there all around us. From the morning till night, we often confront lots of numbers. We are aware of reading and writing numbers now, but what about when numbers were not discovered at all? In those days, counting was done using then available physical objects such as stones, bones, sticks, leaves, and so on. Then they learned to mark lines or curves on rocks, caves, or the pottery they used, and so on.

And then, our counting numbers were introduced. A number is a mathematical concept that is used to express a quantity and used in countings, calculations, or any arithmetic operations and other verticals of mathematics.
For example, \(2,3,0.5, – 6, – 100, – 2.3,\sqrt{2,} \) etc.

The counting numbers are also called numbers, and they include the positive integers from \(1\) to infinity. The set of natural numbers is expressed as \(N\) and \(N = \left\{{1,2,3,4,5……} \right\}\)

There was successor and predecessor for all numbers except \(1\) in natural numbers. Also, there was no way to exhibit the emptiness initially. Thus came the need of showcasing emptiness and zero was introduced. When zero is added to counting numbers, we got whole numbers. Whole numbers are positive numbers from \(0\) to infinity and do not contain any fractional or decimal numbers. The set of whole numbers can be expressed as \(W = \left\{{0,1,2,3,4,5,……} \right\}\)

Introduction of Negative Numbers

We know how to do mathematical operations such as addition, subtraction, multiplication, and division on whole numbers. When we subtract a smaller whole number from a greater whole number, the result is always a whole number. For example, \(30 – 12 = 18,\) the resultant is a whole number. But when we subtract a greater whole number from a smaller whole number, the result is not a whole number.

Therefore, a need was felt to have a set of numbers that includes the numbers with a minus sign. This resulted in the creation of a new set of numbers, which are called negative numbers. When negative numbers were added to whole numbers, we got the set of integers.

The set of Integers can be expressed as \(Z = \left\{{… – 5, – 4, – 3, – 2, – 1,0,1,2,3,4,5,….} \right\}\)

The negative numbers have many applications in real life. For example, we use it to express loss, decrement, depreciation, etc. They are used to denote the temperature (less than zero), in a banking system they are used to represent the overdrawn amount of your account balance. They are also used to show the share price and its ups and downs.

As the usage of numbers could not be confined to integers, more sets of numbers were introduced such as rational numbers, irrational numbers, real numbers, and complex numbers.

1. Rational Numbers: Any number that can be expressed as a ratio of one number to another number is written as rational numbers. It can be written as \(\frac{p}{q}\) form. The symbol \(Q\) represents the rational numbers.
For example, \(\frac{3}{4},\frac{{ – 4}}{5},\frac{{ – 6}}{7},\frac{3}{1},\) etc are the rational numbers.

2. Irrational Numbers: The number that cannot be represented as the ratio of one number over another number, is known as irrational numbers and it is represented by the symbol \(P.\)
For example, \(\sqrt 3 ,\sqrt 2 ,\sqrt 5 , – \sqrt 7 ,\) etc are the irrational numbers.

3. Real Numbers: All the sets of positive and negative integers, fractional and decimal numbers except the imaginary numbers are known as real numbers. It is represented by the symbol \(R.\) For example, \(1.2,3.2222…,4,5,7, – 8, – 8.2,9.1,\) etc are the real numbers.

Representation of Negative Numbers

Negative numbers are represented with a minus \(\left( – \right)\) sign along with the number. These numbers are represented to the left of origin (zero) on a number line, and their values are always less than zero. It can be an integer, decimal number, fraction. They are not present in the set of natural numbers and whole numbers.

For example, \( – 5.2, – 3.7, – 4, – 5, – 7,3.4,\frac{{ – 5}}{6}, – 9.1, – \sqrt 7 \) so on are the negative numbers.

Negative Numbers in Numbers line

A straight line with numbers located at equal intervals along its length is called a number line. It can be stretched infinitely in any direction and is usually presented horizontally.

The numbers can be presented on the number line as,

On a number line, the positive numbers are presented to the right side of zero and negative numbers are presented to the left side of zero.

On the number line, if the numbers move to the right from zero, the value of the numbers raises, and if the numbers move to the left on the number line from zero, the value of the number reduces.

Arithmetic Operations Using Negative Numbers

There are four basic arithmetic operations, they are addition, subtraction, multiplication, and division. It seems difficult when we do these four operations using negative numbers, but it is not the case.

Addition of a Positive and a Negative Number

When a positive and negative number must be added, we should first take the absolute value of the two numbers and find the difference between them. The answer will take the sign of the number which has the larger absolute value.

For example,
1. \(\left({ – 3} \right) + 4\)
Here, the absolute value of \( – 3 = 3\) and the absolute value of \(4 = 4\) So, after getting the difference of absolute values, the answer will take the sign of the number with the bigger absolute value.
Hence, \( – 3 + 4 = 1\)

2. Similarly, \( \left({-\frac{2}{7}} \right) + \left({\frac{5}{7}} \right) = \frac{{ – 2 + 5}}{7} = \frac{3}{7}\)

3. \( – 5 + 1 = – 4\)
Here, the absolute value of \( – 5 = 5\) and the absolute value of \(1 = 1\)
So, the difference of absolute values \( = 5 – 1 = 4\)
The answer will take the sign of the number with the larger absolute value of \( – 5.\)

Similarly, 4. \(\left({ – 1.5} \right) + \left( 1 \right) = \left({ – 0.5} \right)\)

5. \(\left({\frac{3}{7}} \right) + \left({\frac{{ – 5}}{7}} \right) = \frac{{3 – 5}}{7} = \frac{{ – 2}}{7}\)

Addition of Two Negative Numbers

The addition of two negative numbers is done in a similar way to how we add two positive numbers and the only change is in the sign of the answer. When two negative numbers are added, the result will be a negative number. So, when adding two negative numbers, add the absolute value of both the numbers and add a negative sign to the answer.

For example,
1. \(\left({ – 3.5} \right) + \left({ – 2} \right) = \left({ – 5.5} \right)\)
2. \(\left({ – \frac{2}{3}} \right) + \left({ – \frac{5}{3}} \right) = \frac{{ – 2 – 5}}{3} = \frac{{ – 7}}{3}\)
3. \(\left({ – 5} \right) + \left({ – 2} \right) = – 5 – 2 = \left({ – 7} \right)\)

Subtraction of Two Negative Numbers

When two negative numbers are subtracted, then it becomes the addition of a positive and negative number.

For example,
1. \(\left({ – 4} \right) – \left({ – 3} \right) = – 4 + 3 = – 1\)
2. \(\left({ – 4.2} \right) + \left({ – 2.1} \right) = \left({ – 6.3} \right)\)
3. \(\left({ – \frac{1}{4}} \right) + \left({ – \frac{3}{4}} \right) = \frac{{ – 1 – 3}}{4} = \frac{{ – 4}}{4} = -1\)

Subtraction of Positive and Negative Numbers

When a positive and negative number is subtracted, check whether a negative number is to be subtracted from a positive number or a positive number is to be subtracted from a negative number.

If a positive number is subtracted from a negative number, then add both the numbers and give a negative sign for the answer.

For example,
1. \( – 3 – \left( 2 \right) = – 3 – 2 = -5\)
2. \(\left({ – 2.2} \right) – \left({ – 2.1} \right) = – 2.2 + 2.1 = – 0.1\)
3. \(\left({ – \frac{1}{4}} \right) – \left({ – \frac{3}{4}} \right) = \frac{{ – 1 + 3}}{4} = \frac{2}{4} = \frac{1}{2}\)

If a negative number is subtracted from a positive number, then add both the numbers and give a positive sign for the answer.

For example,
1. \(3 – \left({ – 2} \right) = 3 + 2 = 5\)
2. \(\left({2.2} \right) – \left({ – 2.1} \right) = 4.3\)
3. \(\left({\frac{1}{4}} \right) – \left({ – \frac{3}{4}}\right) = \frac{{1 + 3}}{4} = \frac{4}{4} = 1\)

Multiplication of Two Negative Numbers

When two negative numbers are multiplied, the product will be a positive number.

For example,
1. \( – 3 \times – 2 = 6\)
2. \(\left({ – 1.2} \right) \times \left({ – 2} \right) = 2.4\)
3. \(\left({\frac{{ – 1}}{4}} \right) \times \left({\frac{{ – 3}}{4}} \right) = \frac{{\left({ – 1} \right) \times \left({ – 3} \right)}}{4} = \frac{3}{4}\)

Multiplication of Positive and Negative Numbers

When a negative and positive number is multiplied, the product will be a negative number.

For example,
1. \( – 3 \times 2 = – 6\)
2. \(\left({ – 1.2} \right) \times \left( 2 \right) = – 2.4\)
3. \(\left({\frac{{ – 1}}{4}} \right) \times \left({\frac{3}{4}} \right) = \frac{{\left({ – 1} \right) \times \left( 3 \right)}}{4} = \frac{{ – 3}}{4}\)

Division of Two Negative Numbers

When two negative integers are divided, the quotient will be a positive integer.

1. \(\left({ – 3} \right) \div \left({ – 2}\right) = \frac{3}{2} = 1.5\)
2. \(\left({ – 1.2} \right) \div \left({ – 2} \right) = 0.6.\)
3. \(\left({\frac{{ – 1}}{4}} \right) \div \left({\frac{{ – 3}}{4}} \right) = \frac{1}{3}\)

Division of Positive and Negative Numbers

When a positive and negative number is divided, the quotient will be a negative integer.

1. \(\left( 3 \right) \div \left({ – 2} \right) = \frac{{ – 3}}{2} = – 1.5\)
2. \(\left({ – 1.2} \right) \div \left( 2 \right) = – 0.6.\)
3. \(\left({\frac{{ – 1}}{4}} \right) \div \left({\frac{3}{4}} \right) = \frac{{ – 1}}{3}\)

The division of a negative number by zero is undefined.

Solved Examples – Negative Numbers

Q.1. What temperature change will a traveller experience when he reaches from the sea level at \({25^ \circ }{\text{C}}\) to the high altitude, where the temperature is \({-5^ \circ }{\text{C}}\)?
Ans: Temperature at the sea level \( = {25^ \circ }{\text{C}}\)
The temperature at the high altitude \( = {-5^ \circ }{\text{C}}\)
Hence, the difference in temperature \( = {25^ \circ }{\text{C-}}\left({ – {5^ \circ }{\text{C}}} \right)\)
\( = {25^ \circ }{\text{C + }}{{\text{5}}^ \circ }{\text{C}} = {30^ \circ }{\text{C}}\)

Q.2. Subtract \(\left({\frac{{ – 20}}{{30}}} \right) – \left({\frac{{ – 9}}{{30}}} \right).\)
Ans: Given \(\left({\frac{{ – 20}}{{30}}} \right) – \left({\frac{{ – 9}}{{30}}} \right)\)
Since both are negative like fractions,
\( \Rightarrow \frac{{ – 20}}{{30}} – \frac{{ – 9}}{{30}} = \frac{{ – 20 + 9}}{{30}}\)
\( = \frac{{ – 11}}{{30}}.\)

Q.3.Simplify \( – 6 – \frac{{ – 1}}{2}.\)
Ans: Given, \( – 6 – \left({\frac{{ – 1}}{2}} \right)\)
\( – 6,\left({\frac{{ – 1}}{2}} \right)\) both belong to negative numbers
\( – 6 – \left({\frac{{ – 1}}{2}} \right) = – 6 + \frac{1}{2}\)
\( = \frac{{ – 12 + 1}}{2} = \frac{{ – 11}}{2}\)

Q.4. Find the predecessor of the following integers.
a) \( – 100\)
b) \( – 16\)
Ans: Predecessor of a number is obtained by subtracting \(1\) from the number.
a) Predecessor of \( – 100\) is \( – 100 – 1 = – 101\)
b) Predecessor of \( – 16\) is \( – 16 – 1 = – 17\)

Q.5. Multiply \(\left({ – 4} \right)\) and \(\left({ – 1.2} \right).\)
Ans: Given the negative numbers \( – 4\) and \( – 1.2.\)
When we multiply two negative numbers, we will get a positive number as a resultant.
Thus, \( – 4 \times – 1.2 = 4.8\)

Summary

In this article, we have learned the definition of negative numbers, examples, and its properties. We also discussed the arithmetic operations on negative numbers such as addition, subtraction, multiplication, and division.

Frequently Asked Questions (FAQ) – Negative Numbers

Q.1. Explain negative numbers with examples.
Ans: Negative numbers are represented with a minus \(\left( – \right)\) sign along with the number. These numbers are represented to the left of origin (zero) on a number line, and their values are always less than zero. It can be an integer, decimal number, fraction, or so on. They are not present in the set of natural numbers and whole numbers.
For example, \( – 5.2, – 3.7, – 4, – 5, – 7, – 3.4,\frac{{ – 5}}{6},-9.1, – \sqrt 7 \) so on are the negative numbers.

Q.2. What are the rules for the negative numbers?
Ans:
1. When a positive and negative number must be added, we should first take the absolute value of the two numbers and find the difference between them. The answer will take the sign of the number which has the larger absolute value.
2. Adding two negative numbers adds the absolute value of both the numbers and adds a negative sign to the answer.
3. When two negative numbers are subtracted, then it becomes the addition of a positive and negative number.
4. If a positive number is subtracted from a negative number, add both the numbers and give a negative sign.
5. When two negative numbers are multiplied, the product will be a positive number.
6. When a negative and positive number is multiplied, the product will be a negative number.
7. When two negative integers are divided, the quotient will be a positive integer.
8. When a positive and negative number is divided, the quotient will be a negative integer.

Q.3. What is the greatest negative integer?
Ans: Integers can be described as the set of natural numbers and their additive inverse involving zero. So, the highest negative number is \( – 1.\)

Q.4. What is the lowest negative integer?
Ans: If we move away from zero to minus infinity on a number line, the value reduces so, there is no smallest negative number as there are infinite negative numbers.

Q.5. What are the applications of the negative numbers?
Ans: we use the negative numbers to express loss, decrement, depreciation, etc. They are used to denote the temperature (less than zero), in a banking system, they are used to represent the overdrawn amount of your account balance. They are also used to show the share price and its ups and downs.