This entry was posted on December 19, 2021 by Anne Helmenstine (updated on October 3, 2023)

In mathematics, **zero** is both a placeholder digit in numerals and a number with a value of none. Here is a collection of facts about the number zero, a look at its history, and its mathematical rules.

### History

People started using zero (mostly as a placeholder) in Babylon, Central America, and Egypt some time in the 2nd millennium BC. The Egyptians used a hieroglyph for zero by 1770 BC, indicating the base line for pyramid construction. Around the same time, the Babylonians began using a zero symbol as a placeholder. Meanwhile, glyphs from Central America indicate the Olmecs had a zero.

The concept of zero predated its description by many centuries. The Indian astronomer and mathematician Brahmagupta wrote the rules for the mathematics of the number zero in the 7th century (628 AD). The Italian mathematician Fibonacci (Leonardo of Pisa) introduced Hindu-Arabic mathematics to Europe in 1202. Prior to this, Roman numerals were commonly in use, which lacked zero even as a placeholder digit.

### Interesting Number Zero Facts

- As a placeholder, zero helps people tell the difference between numbers that would otherwise look the same. For example, 4 and 40 look the same without zero, even though they have different values. In the number 603, the numeral means there are 6 hundred, no tens, and 3 ones.
- As a number, zero indicates the absence of a value. For example, if you have 2 apples and you eat 2 apples, you have zero apples.
- The first use of “zero” in English was in 1598. The word “zero” comes from the Italian
*zero*, which in turn traces its roots to the Arabic word*ṣifr*, meaning “empty.” - Zero is a number with many other names, including “oh”, nil, nought, naught, ought, aught, cipher, zilch, and zip.
- It also has several symbols, but mostly it appears as a squished circle. The ancient Egyptian hieroglyph of zero or
*nfr*is a heart with a trachea, which also meant “beautiful or good.” The Babylonian zero was two slanted wedges. One Chinese zero (690 AD) was a simple circle, somewhat resembling the open symbol in use today. But, the modern symbol actually comes from the Indian symbol, which was a large dot. - There is no year “zero.” Counting on the calendar goes from 1 BC directly to 1 AD.
- The number zero is even.
- Zero is a whole number.
- It is an integer.
- It is a rational number. In other words, you can express it as the quotient of two integers.
- Zero is a real number. You can draw it on a number line.
- Zero is neither positive nor negative. Although, some types of mathematics consider zero as both positive
*and*negative. - The only number that is both real and imaginary is zero.

### Why Is Zero an Even Number?

Zero is an even number or its **parity** (whether it is even or odd) is even. There are a few rationales for calling zero an even number. The basic reason is because it satisfies the definition of an even number: it is an integer multiple of 2, where 0 x 2 = 0.

There are other reasons, too:

- Zero is divisible by 2 and every multiple of 2. For example, 0 ÷ 2 = 0 and 0 ÷ 4 = 0.
- A decimal integer has the same parity as its last digit. For example, the number 10 is even and its last digit is zero, so 0 is even.
- Numbers on the integer number line alternate between even and odd. The numbers on either side of zero are odd, so 0 is even.
- Zero is the starting point from which natural even numbers are recursively defined.

### What Is the Plural of Zero?

The two plural forms of the word “zero” are “zeros” and “zeroes.” According to *The Oxford Dictionary*, either word is equally fine. However, the word “zeroes” usually finds use when “zero” is a verb. For example, you would say “she zeroes in on the target.” In discussions about the number zero in math, the plural “zeros” is more common.

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### Zero in Math

The number zero has several special properties in math:

#### Zero Addition – Additive Identity

Adding a number plus zero equals that number. This is the additive identity property.

- n + 0 = n
- 2 + 0 = 2
- -5.4 + 0 = -5.4

#### Zero Subtraction

Subtracting zero from a number equals that number.

- n – 0 = n
- 3 – 0 = 3
- -1.75 – 0 = -1.75

Subtracting a number from zero equals the negative value of that number.

- 0 – x = -x
- 0 – 2 = -2
- 0 – (-3) = 3

#### Zero Multiplication

Multiplying a number by zero equals zero.

- n x 0 = 0 x n = 0
- 5 x 0 = 0
- -42 x 0 = 0

#### Zero Division

Zero divided by any non-zero number is zero.

- 0 ÷ x = 0 (providing x is not zero)
- 0 ÷ 8 = 0
- 0 ÷ -12 = 0

A number divided by zero is undefined. This is because 0 lacks a multiplicative inverse. In other words, no real number multiplied by zero equals 1.

- n / 0 = undefined
- 1 / 0 = undefined
- -4 / 0 = undefined

Note that in certain mathematical disciplines, dividing 1 or a positive number by zero is infinity. But, even here, 0/0 is undefined.

#### Zero and Exponents

Raising a number to the zero power equals 1. In other words, when the exponent is zero, it does not matter what the base is. The exception is when that number is zero (in some contexts).

- X
^{0}= 1 (where x is not 0) - 5
^{0}= 1 - -2
^{0}= 1 - 0
^{0}= 1 (usually) - 0
^{0}= undefined (sometimes)

In algebra and combinatorics, 0^{0}= 1. For example, the binomial theorem is only value for x = 0 when 0^{0} = 1. In mathematical analysis and some programming languages, 0^{0} is undefined.

Zero raised to the power of a number equals 0, providing that number is non-zero and positive.

- 0
= 0, when x ≠ 0^{x} - 0
^{5}= 0 - 0
^{–}= undefined^{x} - 0
^{-1}= undefined (basically this is the same as 1 ÷ 0) - 0
^{-2.5}= undefined - 0
^{0}= undefined or 1, depending on the discipline

#### More Math Rules for Zero

- 0! = 1 (zero factorial equals one)
- √0= 0
- log
_{b}(0)is undefined - sin 0º = 0
- cos 0º = 1
- tan 0º = 0
- The sum of 0 numbers (the empty sum) equals zero.
- The product of 0 numbers (the empty sum) is 1.
- The derivative 0′ = 0.
- The integral ∫0 d
*x*= 0 +*C*

### References

- Anderson, Ian (2001).
*A First Course in Discrete Mathematics*. London: Springer. ISBN 978-1-85233-236-5. - Bourbaki, Nicolas (1998).
*Elements of the History of Mathematics*. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3-540-64767-8. - Ifrah, Georges (2000).
*The Universal History of Numbers: From Prehistory to the Invention of the Computer*. Wiley. ISBN 978-0-471-39340-5. - Matson, John (2009). “The Origin of Zero“.
*Scientific American*. Springer Nature. - Soanes, Catherine; Waite, Maurice; Hawker, Sara, eds. (2001).
*The Oxford Dictionary, Thesaurus and Wordpower Guide*(2nd ed.). New York: Oxford University Press. ISBN 978-0-19-860373-3. - Weil, Andre (2012).
*Number Theory for Beginners*. Springer Science & Business Media. ISBN 978-1-4612-9957-8.