A number written with a decimal point, showing parts smaller than 1 as tenths, hundredths and so on.
Precisely: A number written in base ten with digits after a decimal point: 3.75 = 3 + 7/10 + 5/100.
How it works
Each place after the point is a tenth of the place before it. Every fraction becomes a decimal by dividing top by bottom; the decimal either ends (terminates) or repeats for ever.
Examples
3.75 is a decimal: 3 wholes, 7 tenths and 5 hundredths.
1/8 as a decimal is 0.125, and 1/3 is 0.333…, which never ends.
A price of ₹12.50 is a decimal: twelve rupees and fifty paise.
Do not confuse: 0.5 and 0.50 are the same number; 0.5 is bigger than 0.45 even though 45 is bigger than 5.
Used in
money and measurement, calculators, rational and irrational numbers
OpenStax: Intermediate Algebra 2e, 5.1 Key Terms (degree of a polynomial)
denominator
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The bottom number of a fraction: it says how many equal parts the whole is split into.
Precisely: In the fraction p/q, the number q (q ≠ 0), the divisor.
How it works
The bigger the denominator, the smaller each part. To add or subtract fractions, first give them a common denominator (the LCM of the denominators). A denominator can never be 0.
Examples
In 3/8 the denominator is 8: the whole is cut into 8 equal parts.
To add 3/4 and 5/6, use the common denominator 12, the LCM of 4 and 6.
7/20 ends as a decimal (0.35) because its denominator 20 = 2² × 5 has only the primes 2 and 5.
Do not confuse: The denominator is the bottom; the numerator is the top. A memory aid: Denominator is Down.
Used in
fractions, rational numbers, probability (the number of possible outcomes)
A pair of equations that are really the same equation, so they have infinitely many solutions.
Precisely: A pair of linear equations that are equivalent (a₁/a₂ = b₁/b₂ = c₁/c₂): their lines coincide and every point on them is a solution. A dependent pair is always consistent.
How it works
One equation is a multiple of the other. Elimination ends in a statement that is always true, like 0 = 0, which is the sign of a dependent pair.
Examples
x + y = 5 and 2x + 2y = 10 are a dependent pair: every point on x + y = 5 works.
2x + 3y − 9 = 0 and 4x + 6y − 18 = 0 are a dependent pair.
For a dependent pair, subtracting 2 × (x + y = 5) from 2x + 2y = 10 gives 0 = 0.
Do not confuse: Dependent pairs have infinitely many solutions, not none; the pair with none is inconsistent.
Used in
pairs of linear equations, systems of equations (Class 12)
OpenStax: Elementary Algebra 2e, 5.1 Solve Systems of Equations by Graphing (dependent equations)
derivative
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The rate at which a function changes: the slope of its graph at a point. The derivative of y with respect to x is written dy/dx.
Precisely: The derivative of f at x is f′(x) = lim h→0 (f(x + h) − f(x)) ÷ h, when this limit exists; it is the slope of the tangent to y = f(x) at that point.
How it works
Take a chord from (x, f(x)) to a nearby point (x + h, f(x + h)). Its slope is (f(x + h) − f(x)) ÷ h. As h shrinks to 0, the chord turns into the tangent, and its slope becomes the derivative. Rules like the power rule do this once and for all.
Examples
The derivative of x³ is 3x².
Speed is the derivative of distance with respect to time.
The derivative of sin x is cos x.
Where the derivative is 0, the tangent is flat.
Do not confuse: The derivative is a rate at one instant, not an average over a stretch: the chord's slope is the average, the tangent's slope is the derivative.
Used in
slopes of tangents, speed and acceleration, maxima and minima in Class 12
A chord that passes through the centre of a circle; it is the longest chord, and it is twice as long as the radius.
Precisely: A chord of a circle through its centre; its length is 2r, and every diameter is a line of reflection symmetry of the circle.
How it works
Fold a paper circle so that its edges meet: the crease is a diameter. The angle a diameter makes at any point of the circle is 90° (the angle in a semicircle), because its angle at the centre is a straight angle, 180°, and the angle at the circle is half of that.
Examples
A circle of diameter 26 cm has radius 13 cm.
If AB is a diameter and C is any other point on the circle, then ∠ACB = 90°.
Every diameter cuts the circle into two equal halves, called semicircles.
No chord is longer than the diameter: its distance from the centre is 0, the smallest possible.
Do not confuse: Every diameter is a chord, but most chords are not diameters: only those through the centre are.
Used in
measuring round objects, circle theorems, circumference, πd (Class 10)
One square minus another, which always splits as a² − b² = (a + b)(a − b).
Precisely: An expression of the form a² − b²; the identity a² − b² = (a + b)(a − b) factorises it.
How it works
Multiply (a + b)(a − b): the middle terms +ab and −ab cancel, leaving a² − b². Read backwards, it factorises; rewritten as a² = (a + b)(a − b) + b², Śhrīdharāchārya (750 CE) used it to square numbers quickly.
Examples
x² − 9 is a difference of two squares: x² − 9 = (x + 3)(x − 3).
104 × 96 = 100² − 4² = 9984, a difference of two squares.
Using the difference of two squares the other way: 97² = (97 + 3)(97 − 3) + 3² = 9400 + 9 = 9409.
Do not confuse: A sum of two squares like x² + 9 does not factorise this way; the minus sign is essential.
OpenStax: Elementary Algebra 2e, 7.4 Factor Special Products
digit
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One of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 used to write every number.
Precisely: A single symbol of a positional number system; in base ten the digits are 0 to 9, and a number's value comes from its digits and their places.
How it works
Ten digits are enough for any number because each position multiplies by 10. The digits of a number carry information: the last digit decides evenness, the digit sum decides divisibility by 3 and 9.
Examples
7429 has four digits: 7, 4, 2 and 9.
The digit sum of 7524 is 7 + 5 + 2 + 4 = 18, so 7524 is divisible by 9.
The largest three-digit number is 999 and the smallest is 100.
Do not confuse: A digit is a single symbol; a number can have many digits. 45 is a number with the digits 4 and 5.
A fixed straight line used with the focus to define a parabola: every point on the parabola is as far from the focus as from the directrix.
Precisely: A fixed line such that a conic is the set of points whose distance from the focus is e times their distance from the line; for a parabola e = 1, and for y² = 4ax the directrix is x = −a.
How it works
Take any point on y² = 4ax. Its distance to the focus (a, 0) and its distance straight across to the line x = −a come out the same: both are x + a. That equal distance is what makes the curve a parabola.
Examples
The directrix of y² = 8x is the line x = −2.
The vertex of a parabola lies halfway between the focus and the directrix.
For x² = 12y the directrix is y = −3.
Do not confuse: The directrix is a line outside the curve; the axis is the line through the focus at right angles to it.
Used in
defining a parabola, writing its equation from the focus, eccentricity of conics
The number b² − 4ac, which tells you how many real roots ax² + bx + c = 0 has, without solving it.
Precisely: For ax² + bx + c = 0 (a ≠ 0), D = b² − 4ac: D > 0 gives two distinct real roots, D = 0 two equal real roots, and D < 0 no real roots.
How it works
It is the part under the square root in the quadratic formula. A positive number has two square roots (±), zero has one, and a negative number has no real square root, which is why it decides the nature of the roots.
Examples
For 2x² − 4x + 3 = 0 the discriminant is 16 − 24 = −8, so there are no real roots.
For x² − 6x + 9 = 0 the discriminant is 36 − 36 = 0: two equal roots, both 3.
For 2x² − 7x + 3 = 0 the discriminant is 49 − 24 = 25, a perfect square, so the two roots are rational: 3 and 1/2.
Do not confuse: The discriminant is b² − 4ac, not √(b² − 4ac); and it tells how many roots there are, not what they are.
Used in
the nature of roots, whether a parabola meets the x-axis, finding k so that roots are equal
A rule that gives the straight-line distance between two points from their coordinates: √((x₂ − x₁)² + (y₂ − y₁)²).
Precisely: For points (x₁, y₁) and (x₂, y₂), the distance is d = √((x₂ − x₁)² + (y₂ − y₁)²), the length of the segment joining them.
How it works
The horizontal gap and the vertical gap between the two points are the legs of a right triangle. By the Baudhāyana-Pythagoras theorem, the straight distance is its hypotenuse.
Examples
By the distance formula, (1, 2) and (4, 6) are √(3² + 4²) = 5 units apart.
The distance formula shows that (3, 4) is exactly 5 units from the origin.
To check whether a triangle is isosceles, use the distance formula on each of its three sides.
Do not confuse: It gives the straight distance, not the distance travelled along a grid of streets (for (1, 2) to (4, 6) that would be 3 + 4 = 7).
Used in
coordinate geometry, checking shapes: isosceles, right-angled, points on a line, circles on the coordinate plane
OpenStax: College Algebra 2e, 2.1 Key Terms (distance formula)
distributive property
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Multiplying a sum is the same as multiplying each part and then adding: a(b + c) = ab + ac.
Precisely: For all numbers a, b and c, a(b + c) = ab + ac; multiplication distributes over addition (and subtraction).
How it works
Think of a rectangle a wide and b + c long: its area is the two smaller rectangles ab and ac added. Used twice, it multiplies any two brackets, and that is how every identity in the chapter is proved.
Examples
By the distributive property, 7 × 103 = 7 × 100 + 7 × 3 = 721.
Any whole number a divided by a whole number b gives a quotient q and a remainder r smaller than b: a = bq + r.
Precisely: For positive integers a and b there are unique integers q and r with a = bq + r and 0 ≤ r < b; used again and again, it finds the HCF (Euclid's algorithm).
How it works
It is ordinary long division written as an equation. For the HCF of a and b: divide the bigger by the smaller, then divide the smaller by the remainder, and so on; the last non-zero remainder is the HCF.
Examples
By the division algorithm, 17 = 5 × 3 + 2: quotient 3, remainder 2.
Euclid's division algorithm for the HCF of 455 and 42: 455 = 42 × 10 + 35, 42 = 35 × 1 + 7, 35 = 7 × 5 + 0, so the HCF is 7.
The division algorithm says every whole number is 2q or 2q + 1, which is why each one is even or odd.
Do not confuse: The remainder must be smaller than the divisor: 17 = 5 × 2 + 7 is true but is not the division algorithm's answer.
Used in
long division, the HCF by Euclid's algorithm, remainders and clock arithmetic, polynomial division (Class 10 and 11)
A formula for the sine, cosine or tangent of 2A using the ratios of A, such as sin 2A = 2 sin A cos A.
Precisely: An identity expressing a trigonometric ratio of 2A in terms of ratios of A: sin 2A = 2 sin A cos A, cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A, and tan 2A = 2 tan A ÷ (1 − tan²A).
How it works
Put B = A in the compound angle formulas. For example sin(A + A) = sin A cos A + cos A sin A, which is 2 sin A cos A.
Examples
By the double angle formula, sin 60° = 2 sin 30° cos 30° = 2 × ½ × √3/2 = √3/2.
The double angle formula cos 2A = 1 − 2sin²A turns sin²A into (1 − cos 2A)/2.
If tan A = ½, the double angle formula gives tan 2A = 1 ÷ (3/4) = 4/3.
The three forms of the double angle formula for cos 2A each suit a different problem.
Do not confuse: sin 2A is not 2 sin A: sin 60° is about 0.87, while 2 sin 30° is 1.
Used in
simplifying expressions, half-angle values, proving identities, integration in Class 12