Key Uses and Properties of Zero for Students
All the list of whole numbers is the part of integers that starts with zero and ends at infinity. Since we know integers are the numbers that can be produced by adding and subtracting one’s together.
Suppose, if we add 1 + 1, we get 2, which is an integer.
In the same way, if we subtract 1 from 1, we get 0,
1 - 1 = 0
Therefore, 0 is a whole number.
0 as a Number
0 is the integer that precedes the positive number one and follows the number −1. In most of the numerical systems, 0 was identified even before the idea of 'negative integers' was accepted.
Zero is an integer that quantifies a count or an amount of null size; that is, if the number of your brothers is equal to zero, this means the same thing as having no brothers.
And if something has a weight equal to zero, it means it has no weight.
If the difference between the number of pieces in two piles is equal to zero, it means the two piles have an equal number of pieces. Even before counting starts, the result can be assumed to be equal to zero; that is the number of items counted before you count the first item, and counting the first item brings the result equal to one. And if there are no items to be counted, the number zero remains the final result.
While all mathematicians accept 0 as a number, some non-mathematicians would say that zero is not a number, arguing one cannot have 0 of something. Others hold that if suppose you have a bank balance of zero rupees, you have a specific quantity of money in your account, that is namely none. It is that latter view that is accepted by mathematicians.
Almost all historians omit the year zero from the proleptic Gregorian and Julian calendars, but astronomers include it in these same calendars. However, the phrase Year Zero can be used to describe or demonstrate any event considered so significant that it virtually starts a new time reckoning.
Zero as a Decimal Digit
Positional notation without the use of zero (using an empty space in tabular arrangements, or the word kha which means "emptiness") is known to have been in use in India from the 6th century. The earliest certain use of zero as a decimal positional digit date to the 9th century. The glyph for the zero digits was written in the shape of a dot, and consequently called Bindu "dot".
The Number Zero
Zero shows that there is no amount. Example: 6 − 6 equals 0 (the difference between six and six is zero). The number zero is also used as a "placeholder" so we can write a numeral properly.
Example: 502 (five hundred and two) could be mistaken for 52 (fifty-two) without the zero in the tens place.
Zero is a Very Special Number
It is halfway between the number −1 and number +1 on the Mathematics Number Line: Zero is neither a negative number nor a positive number. But zero is an even number.
The Idea of Number Zero
The idea of zero, though natural to us now, was not natural to early humans. if there is nothing to count, how can we count it?
Example: You can count dogs, but you can't count an empty space:
(Images will be uploaded soon)
Zero as a Placeholder
About 3,000 years ago when people needed to tell the difference between numbers like number 4 and the number 40. Without zero, they both look the same!
So zero is now used as a "placeholder": it shows "there is no number at this place", like this:
602 - This means 6 hundred, no tens, and 2 ones.
The Value of Zero
Then people started thinking of zero (0) as an actual number.
Example:
"Let’s say, I had 3 oranges, then I ate all the 3 oranges, now I have zero oranges...!"
Additive Identity
And zero has a special property: when we add it to a number we get that number back, unchanged.
Example:
7 + 0 = 7
Adding 0 to 7 gives the answer 7
Also 0 + 7 = 7
This makes it the Additive Identity, which is just a special way of saying "add 0 and we get the identical (same) number we started with".
Here are some of zero's properties:
Special Properties
FAQs on What Is Zero in Mathematics?
1. What is the mathematical definition of zero (0)?
In mathematics, zero (0) is the integer that represents an empty quantity or the concept of 'nothing'. It is the number that lies between the negative and positive integers on the number line. Crucially, it serves as the additive identity, meaning that when zero is added to any number, the number remains unchanged (e.g., x + 0 = x).
2. What are the basic properties of zero in mathematical operations?
Zero has unique properties when used in arithmetic operations, which are fundamental in the CBSE and NCERT syllabus. Key properties include:
Addition: Any number added to zero remains the same (a + 0 = a).
Subtraction: Subtracting zero from a number does not change it (a - 0 = a). Subtracting a number from zero results in its negative (0 - a = -a).
Multiplication: Any number multiplied by zero equals zero (a × 0 = 0).
Division: Dividing zero by any non-zero number results in zero (0 ÷ a = 0). However, division by zero is undefined.
Exponentiation: Any non-zero number raised to the power of zero is 1 (a⁰ = 1). 0⁰ is typically considered an indeterminate form.
3. Who is credited with the invention of zero?
The concept of zero as a number, and not just a placeholder, was developed in India. While the idea evolved over time, the Indian mathematician Brahmagupta, around the 7th century CE, was the first to formalise rules for operations involving zero. The famous mathematician Aryabhata also used zero as a placeholder in his work centuries earlier, laying crucial groundwork for its development.
4. What type of number is zero?
Zero is classified in several ways within the number system. It is considered:
A whole number, as the set of whole numbers starts from 0 (0, 1, 2, ...).
An integer, as it is part of the set of integers which includes positive numbers, negative numbers, and zero (..., -2, -1, 0, 1, 2, ...).
A rational number, because it can be expressed as a fraction with an integer numerator and a non-zero integer denominator (e.g., 0/1).
A real number, as it is included in the set of all rational and irrational numbers.
5. Is zero considered a positive number, a negative number, or neither?
Zero is unique because it is neither positive nor negative. It acts as the boundary or origin on the number line that separates the positive numbers (to its right) from the negative numbers (to its left). This neutral status is a fundamental concept in understanding number properties and inequalities.
6. How is zero different when used as a number versus a placeholder?
Zero has two distinct but equally important roles in mathematics:
As a number: Zero represents a value, specifically the value of 'nothing' or an empty set. For example, if you have 3 apples and eat all 3, you are left with 0 apples.
As a placeholder: In our place-value system, zero is used to hold a place and give value to other digits. For example, in the number 502, the zero indicates that there are 'no tens'. Without it, the number would be 52, which is a completely different value. The zero's role as a placeholder is crucial for writing and understanding large numbers.
7. Why is division by zero undefined in mathematics?
Division by zero is considered undefined because it creates a logical contradiction. Division is the inverse of multiplication. For example, 10 ÷ 2 = 5 because 5 × 2 = 10. If we were to calculate a value for 10 ÷ 0, let's call it 'x', it would mean that x × 0 = 10. However, any number multiplied by 0 is always 0, never 10. Since no number 'x' can satisfy this equation, the operation is undefined. This rule is a critical foundation for algebra and higher mathematics.
8. In algebra, what does it mean to find the 'zero' of a function or a polynomial?
Finding the 'zero' of a function or polynomial (like those studied in CBSE Class 9 and 10) means finding the input value (usually 'x') that makes the output of the function equal to zero. For example, for the polynomial P(x) = x - 4, the zero is 4 because P(4) = 4 - 4 = 0. Geometrically, the zeros of a function are the x-intercepts—the points where the function's graph crosses the x-axis.
