MATLAB: Improper negative number representation

Question

I have encountered a very strange issue. I was having trouble with an if statement that was comparing two numbers. Although the numbers were apparently identical, the equality was seen as false. I dug a little deeper and I found this extremely simple minimal working example. Why is this happening? This doesn't make any sense!

>> format long
>> A = -6*1e-9
A =
    -6.000000000000001e-09

Answer 1

IEEE double precision floating point cannot represent the number -6e-9 exactly. What you are seeing is the decimal representation of the closest binary floating point bit pattern that IEEE double can represent. This is normal behavior for binary floating point systems. See this link for more discussion:

https://www.mathworks.com/matlabcentral/answers/57444-faq-why-is-0-3-0-2-0-1-not-equal-to-zero?s_tid=ta_ans_results

The "exact" decimal conversion of the binary floating point numbers stored for the numbers in your examples:

>> num2strexact(-1*1e-9)
ans =
    '-1.0000000000000000622815914577798564188970686927859787829220294952392578125e-9'
>> num2strexact(-2*1e-9)
ans =
    '-2.000000000000000124563182915559712837794137385571957565844058990478515625e-9'
>> num2strexact(-3*1e-9)
ans =
    '-3.0000000000000003936399275115964879745433790958486497402191162109375e-9'
>> num2strexact(-4*1e-9)
ans =
    '-4.00000000000000024912636583111942567558827477114391513168811798095703125e-9'
>> num2strexact(-5*1e-9)
ans =
    '-5.0000000000000001046128041506423633766331704464391805231571197509765625e-9'
>> num2strexact(-6*1e-9)
ans =
    '-6.000000000000000787279855023192975949086758191697299480438232421875e-9'
>> num2strexact(-7*1e-9)
ans =
    '-7.00000000000000064276629334271591365013165386699256487190723419189453125e-9'
>> num2strexact(-8*1e-9)
ans =
    '-8.0000000000000004982527316622388513511765495422878302633762359619140625e-9'
>> num2strexact(-9*1e-9)
ans =
    '-9.0000000000000011809197825347894639236301372875459492206573486328125e-9'