Error on summation of three numbers

Hi, When we used this commands on commands windows of Matlab 2013a, we found a problem and we couldn't understand why that happened.

>> a=4.2; b=0.8; c=5;

>> a+b-c

ans = 0

>> a-c+b

ans = 2.2204e-16

>> b-c+a

ans = 0

The question is why on the second step (a-c+b) Matlab reached a number except zero! and how we can be sure that doesn't happen anymore?!

Answer 1

[A Jenkins]

Why is 0.3 - 0.2 - 0.1 (or similar) not equal to zero?

[alireza]

Why? any body knows?

[Jos (10584)]

follow the link in A Jenkins' answer …

Answer 2

[Roger Stafford]

To give you a very brief answer, your computer is using a binary representation of floating point numbers and cannot therefore exactly represent 4.2 and .8 . The resulting round-off errors produce the tiny difference you see here. In other words, round-off errors are causing the addition operation to not be strictly associative. It's the same difficulty a decimal calculator would have adding fractions like 1/3, 1/7, etc. It cannot represent them exactly and therefore makes tiny errors with them.