# Why is the value zero not possible with computers?

• fisico30
In summary: It is relevant for 'normal' computers as it is application dependent.standard Matlab (which the OP uses) uses the IEEE format for calculation, which will result in the observed 'non-zero' zeros.However, the Matlab Symbolic Toolbox uses a bignum format and may well (depending upon the type of the number) store numbers as rationals.
fisico30
Hello Forum,
any number that a computer deals with is just an approximation...in what sense?

Why is the number zero not possible? The number zero is often a very very very small number (10^-19) but never exactly zero. Why?

thanks,
fisico30

I think the issue is with floating point numbers. They consist of a mantissa and exponent. While you can set the mantissa to zeros an exponent of zero sets the number equal to 1. So the only way to get near zero is to have a large negative exponent. That number format cannot precisely represent zero. I would be surprised if there are not ways to get around this.

thank you.
I was doing some MATLAB and looking for zeros but all I could find was very small values, no zeros. The function I was using was supposed to give some zeros but it didn't.

Fisico30

Integral said:
I think the issue is with floating point numbers. They consist of a mantissa and exponent. While you can set the mantissa to zeros an exponent of zero sets the number equal to 1.
No, that's not right.

Conceptually, floating point numbers are {sign} {mantissa} X 2{exponent}. Per this wiki page (http://en.wikipedia.org/wiki/IEEE_754-1985), zero can be represented as either positive zero or negative zero, with sign being either 0 or 1, respectively, mantissa being zero for both, and exponent being zero for both.
Integral said:
So the only way to get near zero is to have a large negative exponent. That number format cannot precisely represent zero. I would be surprised if there are not ways to get around this.

fisico30 said:
thank you.
I was doing some MATLAB and looking for zeros but all I could find was very small values, no zeros. The function I was using was supposed to give some zeros but it didn't.
This isn't a problem of the computer not being able to store zero (which it can), but of the software not being able to calculate precisely enough to find a value for which the function value was exactly zero.

fisico30 said:
any number that a computer deals with is just an approximation...in what sense?
This is not true in general. Computers can store many integer values exactly. As already mentioned in this thread, the problem comes with how non-integer real numbers are stored.
fisico30 said:
Why is the number zero not possible? The number zero is often a very very very small number (10^-19) but never exactly zero. Why?
10^(-19) is NOT zero. It's pretty small, but it isn't zero.

IEEE 754 specifies signed zeroes in its floating point standard.
+0 = -0 = 0, so it is possible for zero to be represented on a computer.

How else would we get "Error: Divide by zero" messages?

fisico30 said:
Hello Forum,
any number that a computer deals with is just an approximation...in what sense?

Why is the number zero not possible? The number zero is often a very very very small number (10^-19) but never exactly zero. Why?

thanks,
fisico30

Mark has pretty much covered it except for pointing out that SOME floating point numbers are represented exactly, NOT approximately as you seem to think, while some ARE represented as approximations.

All integers are represented exactly UP TO THE LIMIT of the computers ability to express integers. For an 8-bit computer (the VERY early kind) that was 256. For a 16-bit computer, that's 65536, and so forth.

Perhaps it will help you understand the "approximation" business if you think of a decimal computer that has the ability to represent 10 significant digits.

How would that computer represent the answer when doing the division 1/3 ? Well, it would be .3333333333 which is only an approximation because of the necessary rounding. Binary computer have the same problem, just in binary.

phinds said:
How would that computer represent the answer when doing the division 1/3?
One option, available on some calculators, and in some libraries for computers, is fractional math, where numbers are are stored as fractions with integer numerators and denomiators. I'm not sure how practical this is versus being a novelty feature found on some calculators.

rcgldr said:
One option, available on some calculators, and in some libraries for computers, is fractional math, where numbers are are stored as fractions with integer numerators and denomiators. I'm not sure how practical this is versus being a novelty feature found on some calculators.

Seems pretty likely that the OP is talking about normal computers, so this is not relevant.

phinds said:
Seems pretty likely that the OP is talking about normal computers, so this is not relevant.

It is relevant for 'normal' computers as it is application dependent. Standard Matlab (which the OP uses) uses the IEEE format for calculation, which will result in the observed 'non-zero' zeros. However, the Matlab Symbolic Toolbox uses a bignum format and may well (depending upon the type of the number) store numbers as rationals.

Mathcad's symbolic processor works in a similar fashion as does Mathematica and at least a few other symbolic applications.

Seems to me that this is a roundoff-error issue. I've seen it a lot myself.

The only rational numbers that can be represented with a finite number of trailing digits in some number-base system are those whose denominators are nonnegative-integer powers of the base. Otherwise one gets an infinitely-repeating finite sequence of digits after some finite number of initial digits. Irrational numbers are even worse. They cannot be represented with a finite number of trailing digits in *any* base.

It's possible for computers to work with integers and floating-point numbers with arbitrarily high numbers of digits, limited only by the available memory. One has to work with arrays of digits, but it's feasible. It's also possible for computers to work with exact rational numbers as pairs of integers. It's easy to write functions that implement the rules for manipulating fractions.

You can find all this in The GNU MP Bignum Library and in various computer-algebra packages.

I implemented rational numbers myself for the C++ version of my semisimple-Lie-algebra software, because I wanted something faster than GMP. I did it as a template class, with a parameter: the type of integer.

NemoReally said:
It is relevant for 'normal' computers as it is application dependent. Standard Matlab (which the OP uses) uses the IEEE format for calculation, which will result in the observed 'non-zero' zeros. However, the Matlab Symbolic Toolbox uses a bignum format and may well (depending upon the type of the number) store numbers as rationals.

Mathcad's symbolic processor works in a similar fashion as does Mathematica and at least a few other symbolic applications.

Fair enough, but I was talking about the computer at the hardware level, as I believe the OP was, not applications.

lpetrich said:
Seems to me that this is a roundoff-error issue ...

Yes, exactly.

## What is the significance of zero not being possible with computers?

Zero is considered to be the starting point for most counting systems, and its absence in computers can lead to confusion and errors in calculations. Understanding why zero is not possible with computers is essential for accurate data processing and analysis.

## Why can't computers handle the concept of zero?

Computers operate based on binary code, which uses a series of 0s and 1s to represent data. Since zero is represented as all 0s in binary code, the computer cannot differentiate between the absence of a value and the value of zero. This can lead to unexpected results in calculations and data storage.

## How does the lack of zero affect calculations and programming?

Without the concept of zero, calculations and programming become more complex and prone to errors. This is because the absence of a zero value can be interpreted as a positive or negative value, leading to incorrect results. It also requires programmers to account for the absence of zero in their code, making it more challenging to write efficient programs.

## Can the concept of zero be simulated or added to computers?

Yes, there are ways to simulate or add the concept of zero to computers, such as using placeholder values or using a different number system, such as ternary or balanced ternary. However, these methods come with their own limitations and complexities, and the concept of zero remains a fundamental challenge in computer science.

## Are there any advantages to not having zero in computers?

While the absence of zero can create challenges, it also has its advantages. For example, it simplifies the hardware and coding required for computers, making them more efficient and cost-effective. Additionally, it forces programmers to think critically and creatively to work around this limitation, leading to innovative solutions and advancements in computer science.

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