Improper negative number representation

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

[Vittorio Picco]

Interesting discovery, this only happens with some numbers but not others. I am even more confused:

>> -1*1e-9
ans =
    -1.000000000000000e-09

>> -2*1e-9
ans =
    -2.000000000000000e-09

>> -3*1e-9
ans =
    -3.000000000000000e-09

>> -4*1e-9
ans =
    -4.000000000000000e-09

>> -5*1e-9
ans =
    -5.000000000000000e-09

>> -6*1e-9
ans =
    -6.000000000000001e-09

>> -7*1e-9
ans =
    -7.000000000000001e-09

>> -8*1e-9
ans =
    -8.000000000000000e-09

>> -9*1e-9
ans =
    -9.000000000000001e-09

[David Hill]

Floating point numbers! There is lots of matlab documentation on floating point numbers, you just need to search for it.

Answer 1

[James Tursa]

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'

[Vittorio Picco]

Thank you very much, makes sense! I'll modify my IF statement from a == b to an abs(a-b)<tolerance

[James Tursa]

Good choice. When working with floating point comparisons it is best to use tolerances unless you know for sure that the numbers being compared are exact integers.