6 ms·

Okay, fine, I agree that sometimes mathematical notation is bad and we are all computer people here, not math people, so we get really scared of mathematical no

Okay, fine, I agree that sometimes mathematical notation is bad and we are all computer people here, not math people, so we get really scared of mathematical notation. But is (-1)^S 1.M 2^(E-127) so bad that it required a whole blog post to explain it? Except for the "1.M" pseudo-notation to explain the mantissa with the implicit on bit, all of those symbols are found in most programming languages we use. I don't think the value of the blog post was explaining the notation. We all knew what operations to perform when we saw it. The value seems to lie more in thinking of the exponent as the offset on the real line and the mantissa as a certain window inside that offset. Personally, though, this still doesn't seem like a huge, deep insight to me, but maybe I'm just way too used to floating point and have forgotten how hard it was to learn this. I did learn about mantissa, exponents, and even learned how to use a log table in high school, but maybe I'm just old and had an unusual high school experience.

kemerover 9y ago
I don't really understand significance of this insight easier. Floating point number is just stored in scientific notation base 2. That's it. And kids learn scientific notation in 7th grade? 8th tops. I mean, swap base 2 to base 10 in this image, and the effect changes from "woah" to "duh, obviously".
coldtea 9y ago
>But is (-1)^S 1.M 2^(E-127) so bad that it required a whole blog post to explain it? Well, for one the expression doesn't tell us anything, could as well describe alien gravity -- even if we know math. You still need to explain what S, M, and E are used for to understand it.
- tpeo 9y ago
  How is this issue unique to strings? Nobody can look at a random graph and know what it stands for either. Otherwise there would be no need for labeled axes. You still need to explain what is it that we're looking at This isn't to say that visualizations aren't useful, of course.
  - coldtea 9y ago
    >How is this issue unique to strings? Who said anything about strings? >Nobody can look at a random graph and know what it stands for either. Well, the parent wrote: "is (-1)^S 1.M 2^(E-127) so bad that it required a whole blog post to explain it?". And the answer is, yes it is. We can look at y = 2*x + 5 and understand it immediately. Sums, limits, and lots of other equations too. Floating point though is an encoding/representation scheme, so it needs more than the mere equation to be understood. I don't just want to know I replace three numbers and do a calculation, I need to know what these numbers are supposed to represent.
    - tpeo 9y ago
      I've mentioned strings, and so did you now. What I don't understand is what's supposed to be uniquely puzzling about strings. Maybe I'm under the influence of the submission, because the guy said he had an allergy for notation. But even after putting a floating point expression in analogy with some geometrical constructs, the whole thing could still be a model of alien gravity, because neither the expression nor the geometric figures carry any inherent meaning in themselves. The only reason why we know that it is in fact just a pure representation of how to approximate real numbers is because the post itself tells us so.
      - chrisoverzero 9y ago
        “Who said anything about {x}?” is an idiomatic equivalent to “Why did you bring up {x} unprompted?” I think you're trying to say that unlabeled formulas are no better or worse than unlabeled graphs and the like, but calling a formula a “string” is muddling it up. (If that is what you meant.) Because we're talking about floating-point data types, people are likely to think of the string data type.
- pharrington 9y ago
  S is sign. M is mantissa. E is exponent. The only term there you might not remember from high school math is mantissa, and a search engine will precisely explain that in two sentences, in terms you do remember from high school math.
  - kllrnohj 9y ago
    > S is sign. M is mantissa. E is exponent. That doesn't actually get you any closer to knowing what the formula means, though. You need to also define that M is unsigned & 23 bits and E is unsigned & 8 bits. And that alone still doesn't get across the resulting limitations like that precision reduces by half every time the exponent bumps. Sure you can reason about it if you devote the brainpower to applying the limitations to the formula, but it's not explained by the formula itself.
    - pharrington 9y ago
      Yes, jordigh did post, mistakenly or deliberately, an incomplete explanation of IEEE754 single precision floats. A number line with alternative labels showing an approximate mantissa and exponent for a given float is certainly one possible way to explain the application of the spec, but obviously, not only does it give less information than the specification, it is not obvious that it is a generally better explanatory tool. For example, it certainly helps Fabien, I assume it helps you and coldtea, but it leaves me completely bewildered (I know what it says, but on it's own, I'm completely lost as to why). edit: jordigh, not coldtea, first brought up the formula in his argument
      - coldtea 9y ago
        >not only does it give less information than the specification That's generally considered a plus (if not a sine qua non) for a simplified explanation, and the larger the specification, the bigger plus. Else, one might as well read the 100s of pages of specs.
        pharrington 9y ago
badsectoracula 9y ago
Some of us are visual thinkers and understand things easier if there are images attached :-). Personally i think almost completely in images so i'd have to somehow visualize something to understand it :-P someone doing it for me helps immensely. Of course i can also think more abstract but it feels as if i'm trying to send megabytes of data to another city through TCP/IP with one byte per send().
- jordigh 9y ago
  Everyone is a visual learner but everyone can also understand a simple arithmetic expression. Everyone thinks almost completely in images. The only difference is that mathematicians have had more practice learning how to transform expressions into visual images. It's a bit like looking at sheet music and being able to hum it in your head. Learning how to read sheet music isn't something that shouldn't even be attempted because you're an "auditory learner". I also think that spending a little extra effort to learn how to read an arithmetic expression is worthwhile and a skill we should foster in society.
  - thethirdone 9y ago
    >Everyone is a visual learner but everyone can also understand a simple arithmetic expression. Everyone thinks almost completely in images. I think it is hard to separate your experience from the collective here. Much of the time I am working on math or programming, I don't think in images; I am just manipulating symbols in a patterned fashion that I have already convinced myself is sound (in a logic sense). I definitely have never thought about floating point as anything other than binary scientific notation. I am quite comfortable with the understanding of exponent, significand, and sign.
    - jordigh 9y ago
      You've never drawn a linked list? A Venn diagram to think about your logical sets? A Hasse diagram for showing set inclusion? A function to see its growth? Never visualised numbers as being on a real line, with in-betweenness properties? The visual cortex is one of the most developed part of our brains. Even blind people use it for mathematics: https://www.newscientist.com/article/2106215-blind-people-use-brains-visual-cortex-to-help-do-maths/ https://www.newscientist.com/article/2106215-blind-people-us... Even when you're doing symbol pushing "blindly", you're still using visual skills to place symbols, different fonts, you align variables, superscripts, subscripts. There's a certain geometry to mathematical notation itself.
JadeNB 9y ago
> But is (-1)^S 1.M 2^(E-127) so bad that it required a whole blog post to explain it? What is the value of this question? If you don't need a blog post to understand it, then don't read it; no harm was done. Obviously the author felt that there was enough potential for confusion that an explanation might be needed, and other commenters here have appreciated it. That's good for some people and harm for no-one, which seems a pretty clear net win.
- jordigh 9y ago
  Because I get frustrated with people. I feel like they refuse to even attempt to understand. They say to themselves "I'm a visual learner" (everyone is) or they seek solace in groups when mathematical notation is difficult to understand. It's ok if I don't understand this, and I don't even have to try, because lots of other people don't get it either! If even an idiot like me can understand mathematical notation, anyone can. In particular, if you're able to understand code, you can understand mathematics. I get frustrated with the mental barriers people set up for themselves when learning mathematics. I think it may even be cultural. I have heard that in China, for example, they don't have this aversion to anything that looks mathematical.
  - kllrnohj 9y ago
    (-1)^S 1.M 2^(E-127) is the concept not the implementation and the implementation is the important part, especially as a programmer. The blog is mostly about the details. How the bits are allocated, how they influence precision, etc... And exactly none of that is included in "(-1)^S 1.M 2^(E-127)"
    - nv-vn 9y ago
      But that's not what the blog post is about. Every other explanation of floating point numbers describes how bits are allocated. The visual part of this article is a drawing of the window and offset thing. Which in this case, is exactly what the expression is saying.
  - dahart 9y ago
1_player 9y ago
The value of the post is that, yes, I could understand the mathematical notation, but I never had the need to dive into how floating point numbers actually work. Now it's on HN in an accessible format, and now I know how IEEE 754 works, and I've made an intuitive mental model of it thanks to Fabien's great teaching skills. Math notation is a compact representation of a concept, not a way to _explain_ it. There's no need to be condescending.

kemerover 9y ago

I don't really understand significance of this insight easier. Floating point number is just stored in scientific notation base 2. That's it. And kids learn scientific notation in 7th grade? 8th tops. I mean, swap base 2 to base 10 in this image, and the effect changes from "woah" to "duh, obviously".

coldtea 9y ago

>But is (-1)^S 1.M 2^(E-127) so bad that it required a whole blog post to explain it? Well, for one the expression doesn't tell us anything, could as well describe alien gravity -- even if we know math. You still need to explain what S, M, and E are used for to understand it.

tpeo 9y ago
How is this issue unique to strings? Nobody can look at a random graph and know what it stands for either. Otherwise there would be no need for labeled axes. You still need to explain what is it that we're looking at This isn't to say that visualizations aren't useful, of course.
- coldtea 9y ago
  >How is this issue unique to strings? Who said anything about strings? >Nobody can look at a random graph and know what it stands for either. Well, the parent wrote: "is (-1)^S 1.M 2^(E-127) so bad that it required a whole blog post to explain it?". And the answer is, yes it is. We can look at y = 2*x + 5 and understand it immediately. Sums, limits, and lots of other equations too. Floating point though is an encoding/representation scheme, so it needs more than the mere equation to be understood. I don't just want to know I replace three numbers and do a calculation, I need to know what these numbers are supposed to represent.
  - tpeo 9y ago
    I've mentioned strings, and so did you now. What I don't understand is what's supposed to be uniquely puzzling about strings. Maybe I'm under the influence of the submission, because the guy said he had an allergy for notation. But even after putting a floating point expression in analogy with some geometrical constructs, the whole thing could still be a model of alien gravity, because neither the expression nor the geometric figures carry any inherent meaning in themselves. The only reason why we know that it is in fact just a pure representation of how to approximate real numbers is because the post itself tells us so.
    - chrisoverzero 9y ago
      “Who said anything about {x}?” is an idiomatic equivalent to “Why did you bring up {x} unprompted?” I think you're trying to say that unlabeled formulas are no better or worse than unlabeled graphs and the like, but calling a formula a “string” is muddling it up. (If that is what you meant.) Because we're talking about floating-point data types, people are likely to think of the string data type.
pharrington 9y ago
S is sign. M is mantissa. E is exponent. The only term there you might not remember from high school math is mantissa, and a search engine will precisely explain that in two sentences, in terms you do remember from high school math.
- kllrnohj 9y ago
  > S is sign. M is mantissa. E is exponent. That doesn't actually get you any closer to knowing what the formula means, though. You need to also define that M is unsigned & 23 bits and E is unsigned & 8 bits. And that alone still doesn't get across the resulting limitations like that precision reduces by half every time the exponent bumps. Sure you can reason about it if you devote the brainpower to applying the limitations to the formula, but it's not explained by the formula itself.
  - pharrington 9y ago
    Yes, jordigh did post, mistakenly or deliberately, an incomplete explanation of IEEE754 single precision floats. A number line with alternative labels showing an approximate mantissa and exponent for a given float is certainly one possible way to explain the application of the spec, but obviously, not only does it give less information than the specification, it is not obvious that it is a generally better explanatory tool. For example, it certainly helps Fabien, I assume it helps you and coldtea, but it leaves me completely bewildered (I know what it says, but on it's own, I'm completely lost as to why). edit: jordigh, not coldtea, first brought up the formula in his argument
    - coldtea 9y ago
      >not only does it give less information than the specification That's generally considered a plus (if not a sine qua non) for a simplified explanation, and the larger the specification, the bigger plus. Else, one might as well read the 100s of pages of specs.
      - pharrington 9y ago

badsectoracula 9y ago

Some of us are visual thinkers and understand things easier if there are images attached :-). Personally i think almost completely in images so i'd have to somehow visualize something to understand it :-P someone doing it for me helps immensely. Of course i can also think more abstract but it feels as if i'm trying to send megabytes of data to another city through TCP/IP with one byte per send().

jordigh 9y ago
Everyone is a visual learner but everyone can also understand a simple arithmetic expression. Everyone thinks almost completely in images. The only difference is that mathematicians have had more practice learning how to transform expressions into visual images. It's a bit like looking at sheet music and being able to hum it in your head. Learning how to read sheet music isn't something that shouldn't even be attempted because you're an "auditory learner". I also think that spending a little extra effort to learn how to read an arithmetic expression is worthwhile and a skill we should foster in society.
- thethirdone 9y ago
  >Everyone is a visual learner but everyone can also understand a simple arithmetic expression. Everyone thinks almost completely in images. I think it is hard to separate your experience from the collective here. Much of the time I am working on math or programming, I don't think in images; I am just manipulating symbols in a patterned fashion that I have already convinced myself is sound (in a logic sense). I definitely have never thought about floating point as anything other than binary scientific notation. I am quite comfortable with the understanding of exponent, significand, and sign.
  - jordigh 9y ago
    You've never drawn a linked list? A Venn diagram to think about your logical sets? A Hasse diagram for showing set inclusion? A function to see its growth? Never visualised numbers as being on a real line, with in-betweenness properties? The visual cortex is one of the most developed part of our brains. Even blind people use it for mathematics: https://www.newscientist.com/article/2106215-blind-people-use-brains-visual-cortex-to-help-do-maths/ https://www.newscientist.com/article/2106215-blind-people-us... Even when you're doing symbol pushing "blindly", you're still using visual skills to place symbols, different fonts, you align variables, superscripts, subscripts. There's a certain geometry to mathematical notation itself.

JadeNB 9y ago

> But is (-1)^S 1.M 2^(E-127) so bad that it required a whole blog post to explain it? What is the value of this question? If you don't need a blog post to understand it, then don't read it; no harm was done. Obviously the author felt that there was enough potential for confusion that an explanation might be needed, and other commenters here have appreciated it. That's good for some people and harm for no-one, which seems a pretty clear net win.

jordigh 9y ago
Because I get frustrated with people. I feel like they refuse to even attempt to understand. They say to themselves "I'm a visual learner" (everyone is) or they seek solace in groups when mathematical notation is difficult to understand. It's ok if I don't understand this, and I don't even have to try, because lots of other people don't get it either! If even an idiot like me can understand mathematical notation, anyone can. In particular, if you're able to understand code, you can understand mathematics. I get frustrated with the mental barriers people set up for themselves when learning mathematics. I think it may even be cultural. I have heard that in China, for example, they don't have this aversion to anything that looks mathematical.
- kllrnohj 9y ago
  (-1)^S 1.M 2^(E-127) is the concept not the implementation and the implementation is the important part, especially as a programmer. The blog is mostly about the details. How the bits are allocated, how they influence precision, etc... And exactly none of that is included in "(-1)^S 1.M 2^(E-127)"
  - nv-vn 9y ago
    But that's not what the blog post is about. Every other explanation of floating point numbers describes how bits are allocated. The visual part of this article is a drawing of the window and offset thing. Which in this case, is exactly what the expression is saying.
- dahart 9y ago

1_player 9y ago

The value of the post is that, yes, I could understand the mathematical notation, but I never had the need to dive into how floating point numbers actually work. Now it's on HN in an accessible format, and now I know how IEEE 754 works, and I've made an intuitive mental model of it thanks to Fabien's great teaching skills. Math notation is a compact representation of a concept, not a way to _explain_ it. There's no need to be condescending.