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math tricks

Mesh analysis, or why that matrix is a picture of the circuit Every entry is a branch you can point at. A diagonal is the sum of the resistors around one loop; an off-diagonal is minus the resistor two loops share, negative because the two currents run through it in opposite directions. Only the loop holding the source gets a voltage, so V = [16, 0, 0] and the whole circuit comes down to three numbers. Written with Manim in scripts/mesh_analysis.py, after a reel by MAE Academy; the circuit and the numbers are theirs, the drawing is not.
A handwritten and hand-drawn sheet titled "Months Days Trick (Knuckle Method)". A pencil drawing of a closed fist runs across the middle, its four knuckles and the three dips between them labelled Jan to Jul with 31 on each knuckle, 30 in each dip, and 28/29 written under February; an arrow curls back to the first knuckle beside the note "start again from July for remaining months". Underneath, a table gives all twelve months and their day counts, and two boxes restate the pattern 31 - 28/29 - 31 - 30 - 31 - 30 - 31, then repeat.
The knuckle method for how many days are in a month Make a fist and count across the knuckles from January: a knuckle is 31 days, the dip between two is 30. Seven months takes you to July on the last knuckle, so start over at the first one for August and the pattern carries to December. February is the exception and is 28, or 29 in a leap year.
A ruled notebook page in blue ink, headed "Vedic Maths Trick - Multiply Two Numbers Near 100 (Base 100 Trick)". A three-line rule at the top is followed by three worked examples -- 96 x 98 = 9408, 103 x 107 = 11021 and 94 x 97 = 9118 -- each written out in steps on the left and drawn on the right as a cross: the two numbers above, their differences from 100 beside them, and arrows crossing down to the left and right halves of the answer. A boxed shortcut at the foot restates it in two lines.
Multiplying two numbers near 100, the base-100 trick Write each number as 100 plus or minus something. The left part is either number cross-adjusted by the other's difference -- 96 - 2, or equally 98 - 4, both 94. The right part is the two differences multiplied, written in two digits: 4 x 2 = 08. So 96 x 98 = 9408, and 103 x 107 = 11021 the same way, carrying the extra digit left when the right part runs over.
A typeset note headed "Gradient : The Direction Nature Chooses". It defines the gradient of a scalar field phi as the sum of its partial derivatives along x, y and z with unit vectors on them, factored into the del operator applied to phi, with red and blue annotations saying that each component gives the rate of change along that axis and that the gradient of a scalar produces a vector. Below is a shaded three-dimensional surface of hills drawn with contour lines: a point P on a slope carries a long red arrow up the steepest way to a peak and a short green arrow pointing down the slope, where a red ball rests.
The gradient of a scalar field, and why it points the way it does Take the partial derivative of a scalar field along each axis and hang the unit vectors on them, and what falls out is a vector: grad phi. Each component says how fast the field changes along that axis, so the vector as a whole points the way the field climbs fastest -- perpendicular to the contour lines, its length the rate of that climb. Downhill is the same vector with a minus sign, which is the direction anything left alone on the slope actually takes.
A ruled notebook page in blue ink headed "Vedic Maths Finger Trick - Table of 8 (Finger Multiplication Trick)". A five-line rule at the top is followed by three worked examples -- 8 x 6 = 48, 8 x 7 = 56 and 8 x 9 = 72 -- each with a pair of drawn hands beside it, fingers numbered 6 to 10 from the little finger, showing which two fingers touch and which are counted as tens and which as ones. A boxed shortcut at the foot restates it in a line.
The eight times table on your fingers Number both hands from little finger to thumb as 6, 7, 8, 9, 10. To multiply any two of those, touch the two fingers together. The touching fingers and every finger below them are tens; the fingers left above the join on each hand are multiplied together for the ones. So 8 x 7 touches four fingers and below -- fifty -- with 2 and 3 left above, giving 56.
A ruled notebook page in blue ink headed "Vedic Maths Trick - Multiply Numbers Whose Last Digits Add to 10 (Same Tens Digit Trick)". A three-line rule at the top sits beside a boxed schematic of the method written as a b x a c giving a(a+1) alongside b x c. Three worked examples follow -- 43 x 47 = 2021, 62 x 68 = 4216 and 84 x 86 = 7224 -- each broken into the same four steps, with a boxed shortcut at the foot.
Multiplying two numbers with the same tens digit whose last digits add to ten It works only in that case, and then it is two multiplications. The left part is the tens digit times the next number up: for 43 x 47, that is 4 x 5 = 20. The right part is the last digits multiplied, written in two digits: 3 x 7 = 21. Put them together and 43 x 47 = 2021.