Modular Arithmetic vs Touch Trigonometry

A calculator for arithmetic modulo N. It lets you choose a fixed modulus, and then make lots of calculations without having to press a "mod" button again and again. It also: - follows the order convention; - supports arbitrarily large numbers; - performs fast modular division and exponentiation; - can show a full transcript of your calculation. Modular arithmetic is a "calculus of remainders". It features throughout mathematics and computer science, and has applications from cryptography to barcodes to music. The basic idea is that you choose a modulus N, and then reduce every number to one of the integers 0,1,2,...,N−1 according to what remainder it leaves when dividing by N. For example, using a modulus of 17: 40 ≡ 6 (since 40 ÷ 17 leaves a remainder of 6); 17 ≡ 0 (since 17 ÷ 17 leaves no remainder at all). Arithmetic follows these same rules. Still using a modulus of 17: 15 + 7 ≡ 5 (since 22 ≡ 5); 3 × 9 ≡ 10 (since 27 ≡ 10); 5 ^ 3 ≡ 6 (since 125 ≡ 6). Subtraction and division behave in a way that complements addition and multiplication: −1 ≡ 16 (since 16 + 1 = 17 ≡ 0); 1/2 ≡ 9 (since 9 × 2 = 18 ≡ 1); 4 - 7 ≡ 14 (since 14 + 7 = 21 ≡ 4); 7 ÷ 3 = 8 (since 8 × 3 = 24 ≡ 7). There are no negative numbers or fractions: like −1 and 7 ÷ 3 in the examples above, these are also reduced to one of 0,1,...,N−1. As usual, you cannot divide by zero. You also cannot divide if the right hand side has any common factors with the modulus. If we change our modulus to 10, then the following operations all generate errors: 3 ÷ 20 (since 20 ≡ 0); 7 ÷ 8 (since 8 and 10 have a common factor of 2). Integers can be arbitrarily large. For instance, if we set our modulus to 2305843009213693951 (a Mersenne prime), then: 5 ^ 2305843009213693950 ≡ 1 (by Fermat's little theorem). The code is written carefully, and is backed up by a thorough suite of 186 automated tests. This app supports external keyboards, Siri Shortcuts, and (on iPad) Slide Over, Split View, and multiple windows.

Finally understand trigonometry. Touch Trigonometry is the most intuitive way to learn the trigonometric functions. Instead of memorizing formulas, you'll see how sin, cos, tan, and the rest actually work—by touching and exploring a live unit circle and graph. Drag your finger around the circle and watch trig functions respond in real time. You'll be amazed at how quickly the "true meaning" of trigonometry becomes clear. Watch the graph on the right trace their values. In seconds, concepts that confused you for years will suddenly click. This app is great for students struggling with trigonometry or anyone looking for a refresher. HOW IT WORKS On the left: an interactive unit circle. Touch anywhere and drag to change the angle θ (theta). A triangle forms showing the relationships between the angle and the trig values. On the right: live graphs of all six trig functions. As you move the angle, dots trace along each curve showing the current value. Touch the graph side to control the angle from there too. It's bidirectional, immediate, and intuitive. No tutorial needed—just touch and explore. Users have told us that they never understood *why* the trig functions existed—until they used this app. From students to the math curious, Touch Trigonometry offers a truly intuitive way to understand one of the trickiest parts of geometry—at your fingertips! FEATURES • Interactive unit circle with touch control • All 6 trig functions: sin, cos, tan, sec, csc, cot • Live function graphs with real-time value tracking • Bidirectional control (circle or graph) • Right triangle overlay showing opposite/adjacent/hypotenuse • Reference angle markers (30°, 45°, 60°, 90°, etc.) • Switch between degrees and radians • Snap-to-reference-angles mode • Pinch to zoom the graph • Colorblind-friendly color palette • Clean dark theme, no distractions • No ads, no account required WHO IT'S FOR • High school & college students learning trig • Teachers demonstrating concepts in class • Engineers & scientists refreshing fundamentals • Anyone who was told to "just memorize SOHCAHTOA" • Visual learners who need to see it to get it FREE FEATURES • Full unit circle interaction • Sine visualization • Triangle overlay • Degree/radian switching • All UI customization options UNLOCK EVERYTHING (One-Time Purchase) • One-time purchase for lifetime access to all paid features • Cosine, tangent, secant, cosecant, cotangent • All future paid features