https://foundingfuture.com/mathematics/fundamentals/ Recent content in Fundamentals on FoundingFuture Hugo en Mon, 07 Sep 2026 23:58:15 +0200 https://foundingfuture.com/mathematics/fundamentals/numbers-and-sets/ Mon, 31 Aug 2026 20:10:00 +0200 https://foundingfuture.com/mathematics/fundamentals/numbers-and-sets/ <p>Each kind of number was invented because an operation ran out of answers. Subtraction ran out of the counting numbers, division ran out of the whole numbers, and roots ran out of the fractions. Complex numbers are the last of those widenings, and they arrive the same way the earlier ones did.</p> https://foundingfuture.com/mathematics/fundamentals/angles-and-polar-form/ Mon, 31 Aug 2026 21:03:19 +0200 https://foundingfuture.com/mathematics/fundamentals/angles-and-polar-form/ <p>A point of the plane can be named by two coordinates, or by one distance and one turn. The second naming is the polar form, and it is the one under which multiplication of complex numbers becomes easy to read. Reaching it takes an angle measure and two ratios.</p> https://foundingfuture.com/mathematics/fundamentals/matrices/ Mon, 31 Aug 2026 22:25:15 +0200 https://foundingfuture.com/mathematics/fundamentals/matrices/ <p>A vector is a column of numbers. A matrix is a rectangle of them, so a vector is the case of a single column. Adding either works one position at a time. Multiplying two matrices builds each entry of the answer from a whole row and a whole column. That one rule is what the rest of the subject rests on.</p> https://foundingfuture.com/mathematics/fundamentals/iteration-and-orbits/ Mon, 31 Aug 2026 22:05:53 +0200 https://foundingfuture.com/mathematics/fundamentals/iteration-and-orbits/ <p>Applying one rule repeatedly, with each answer becoming the next question, produces a sequence. Which sequence depends entirely on where the process starts. Two starting points a hair apart can produce sequences that end in different places.</p> https://foundingfuture.com/mathematics/fundamentals/signals-and-samples/ Sat, 05 Sep 2026 00:35:37 +0200 https://foundingfuture.com/mathematics/fundamentals/signals-and-samples/ <p>A sound is a pressure that rises and falls in time. A computer cannot hold a curve, only numbers, so it keeps the pressure at regular instants and nothing in between. That list of numbers is a digital signal, and everything an audio program does is arithmetic on the list.</p> https://foundingfuture.com/mathematics/fundamentals/decibels-and-decay/ Sat, 05 Sep 2026 00:35:37 +0200 https://foundingfuture.com/mathematics/fundamentals/decibels-and-decay/ <p>Two sounds that differ by the same ratio sound the same distance apart, however loud they are. A scale for loudness therefore counts ratios rather than differences, and the decibel is that scale. Decay is what one ratio does when it is applied again and again.</p> https://foundingfuture.com/mathematics/fundamentals/filters/ Sat, 05 Sep 2026 00:35:37 +0200 https://foundingfuture.com/mathematics/fundamentals/filters/ <p>A filter is a rule that treats fast changes in a signal differently from slow ones. The simplest rule that does so keeps one number in memory and moves it a fixed fraction of the way towards each new sample. Everything on this page is that rule and its mirror image. The last of them rearranges a delay so that it moves a click in time and leaves every steady tone at its level.</p> https://foundingfuture.com/mathematics/fundamentals/interpolation/ Sat, 05 Sep 2026 00:35:37 +0200 https://foundingfuture.com/mathematics/fundamentals/interpolation/ <p>Given two values and a fraction, interpolation names a value in between. Which value depends on what the two are: numbers on a line, samples of a curve, or directions. Each case has its own rule, and the wrong rule gives an answer that lies between the inputs and is still wrong.</p> https://foundingfuture.com/mathematics/fundamentals/noise/ Sat, 05 Sep 2026 00:35:37 +0200 https://foundingfuture.com/mathematics/fundamentals/noise/ <p>A computer has no dice. What it has is a rule that scrambles a number so thoroughly that the sequence it produces passes for chance. It also has a way of turning that sequence into a signal with a chosen colour.</p> https://foundingfuture.com/mathematics/fundamentals/colour-spaces/ Mon, 07 Sep 2026 23:58:15 +0200 https://foundingfuture.com/mathematics/fundamentals/colour-spaces/ <!-- requires theme shortcode: lead --> <div class="lead"><h2>Three ways to take a colour halfway between two others, and what each one costs.</h2></div> <p>A colour halfway between two others depends on the space the halving happens in. Averaging the bytes a file stores gives one colour. Averaging the light those bytes stand for gives another, and averaging in a space built to match perception gives a third. This page defines the three spaces, gives the transforms between them, and measures the midpoint each produces for one pair of colours.</p>