Another name for the x-coordinate: the first number of a point (x, y).
Precisely: The signed distance of a point from the y-axis, measured along the x-axis.
How it works
Read the first number of the pair. Positive means the point is to the right of the y-axis, negative means to the left, and 0 means it is on the y-axis.
Examples
The abscissa of (7, −2) is 7, so the point is 7 steps to the right.
Every point on the y-axis has abscissa 0.
A point with a negative abscissa lies to the left of the y-axis.
Do not confuse: The abscissa is the first coordinate (x) and the ordinate the second (y). A comes before O in the alphabet, just as x comes before y.
How far a number is from 0 on the number line, without caring which side: |−3| = 3 and |3| = 3.
Precisely: For a real number a, |a| = a when a ≥ 0 and |a| = −a when a < 0; it is never negative.
How it works
Drop the sign. Because it measures a distance, the gap between two numbers a and b on the line is |a − b|, whichever is bigger.
Examples
The absolute value of −7 is 7, because −7 is 7 steps from 0.
The distance between −4 and 3 on the number line is the absolute value |−4 − 3| = 7.
A temperature change from 5 °C to −2 °C has absolute value 7 degrees.
Do not confuse: The absolute value is never negative, but it is not "the number without its minus sign" for expressions: |x − 5| is 5 − x when x is less than 5.
Used in
distance on the number line, the distance between points on a line (Ch 1), inequalities (Class 11)
Flat pieces whose areas stand for x², x and 1, used to build rectangles that show how expressions multiply and factorise.
Precisely: A set of manipulatives: an x²-tile (an x by x square), x-tiles (x by 1 strips) and unit tiles (1 by 1 squares), arranged into a rectangle whose sides are the factors.
How it works
Lay out the tiles for an expression like x² + 7x + 12 and arrange them into one rectangle. The length and the breadth of that rectangle are the two factors, (x + 3) and (x + 4).
Examples
With algebra tiles, x² + 7x + 12 is one x²-tile, seven x-tiles and twelve unit tiles.
Arranged into a rectangle, those algebra tiles have sides x + 3 and x + 4.
Algebra tiles for (2x + 3)(3x + 1) make six x²-tiles, eleven x-tiles and three unit tiles: 6x² + 11x + 3.
Do not confuse: The tiles are a picture of the algebra, not a proof for every case: negative terms and big numbers are easier with the identity itself.
Used in
seeing products of binomials, factorising quadratic expressions, splitting the middle term
A maths phrase made of numbers, letters and operations, like 3x + 5 or a² − 2ab, with no equals sign.
Precisely: A combination of constants and variables joined by addition, subtraction, multiplication, division and powers.
How it works
Its parts joined by + or − are its terms. Put numbers in for the letters and it gives one number; it does not say that anything is equal to anything, so there is nothing to solve.
Examples
5x² + 20xy + 4y² is an algebraic expression with three terms.
The area of a rectangle with sides x + 3 and x + 4 is the algebraic expression (x + 3)(x + 4).
When x = 2, the algebraic expression x² + 4x + 4 has the value 16.
Do not confuse: An expression has no equals sign (3x + 5); an equation says two expressions are equal (3x + 5 = 11).
Used in
writing rules and patterns, identities and factorisation, every chapter of algebra
An equation that stays true whatever numbers you put in for the letters, like (a + b)² = a² + 2ab + b².
Precisely: An equality between two algebraic expressions that holds for every value of the variables in it.
How it works
An ordinary equation is true only for some values (x² − 1 = 24 only for x = 5 or −5). An identity is true for all of them, and you prove it once for all by the distributive property, or see it by cutting a square or a cube into pieces.
Examples
(a + b)² = a² + 2ab + b² is an algebraic identity: try a = −2, b = −3 and both sides give 25.
The amount of turn between two lines that meet at a point, measured in degrees.
Precisely: The figure formed by two rays with a common end point (the vertex); its measure is the amount of rotation from one ray to the other, with a full turn = 360°.
How it works
Measure with a protractor. By size: acute (less than 90°), right (90°), obtuse (between 90° and 180°), straight (180°), reflex (more than 180°). The angles of a triangle add up to 180°.
Examples
A right angle is 90°, a quarter turn, like the corner of a page.
In ∠ABC the vertex of the angle is B, the middle letter.
A triangle with angles 70° and 60° has a third angle of 180° − 130° = 50°.
Do not confuse: The size of an angle does not depend on how long its arms are drawn: a longer drawing of 30° is still 30°.
Used in
geometry everywhere, trigonometry, clocks and directions, circles (angles at the centre)
OpenStax: Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
angle of depression
ANG-gul uv di-PRESH-unClass 10
How far down you tilt your eyes from the horizontal to look at something below you.
Precisely: The angle between the line of sight and the horizontal, when the point viewed is below the horizontal level of the eye.
How it works
The horizontal at the observer is parallel to the ground, so the angle of depression from A to B equals the angle of elevation from B to A (alternate angles). Use whichever right triangle is easier.
Examples
From a bridge the angles of depression of the two banks are 30° and 45°.
From the top of a tall building, the angles of depression of the top and foot of an 8 m building are 30° and 45°; the tall building is 4(3 + √3) ≈ 18.93 m high.
A girl on a cliff sees a boat at an angle of depression of 30°; from the boat, the angle of elevation of the girl is also 30°.
Do not confuse: The angle of depression is inside the triangle's top corner only after using alternate angles; it is measured down from the horizontal, not from the vertical wall.
Used in
heights and distances, lighthouses and ships, aircraft and runways
OpenStax: Algebra and Trigonometry 2e, 7.2 Right Triangle Trigonometry (angle of depression)
angle of elevation
ANG-gul uv el-uh-VAY-shunClass 10
How far up you tilt your eyes from the horizontal to look at something above you.
Precisely: The angle between the line of sight and the horizontal, when the point viewed is above the horizontal level of the eye.
How it works
The object, your eye and the point straight below the object make a right triangle. With the distance along the ground and the angle, tan gives the height above eye level.
Examples
From 15 m away the angle of elevation of a tower's top is 60°, so the tower is 15 tan 60° = 15√3 ≈ 25.98 m tall.
An observer 1.5 m tall, 28.5 m from a chimney, sees its top at an angle of elevation of 45°: the chimney is 28.5 + 1.5 = 30 m.
A tower's shadow is 40 m longer when the Sun's angle of elevation is 30° than when it is 60°, so the tower is 20√3 ≈ 34.64 m tall.
Do not confuse: It is measured from the horizontal, not from the vertical. Looking up gives elevation; looking down gives depression.
Used in
heights and distances, surveying, the Sun's altitude and shadows
OpenStax: Algebra and Trigonometry 2e, 7.2 Right Triangle Trigonometry (angle of elevation)
approximation
uh-prok-si-MAY-shunFoundation
A value that is close to the true value but not exactly equal to it.
Precisely: A number used in place of an exact value, within some known error; written with the sign ≈, "is approximately equal to".
How it works
Choose an approximation good enough for the job: 3.14 or 22/7 for π in school sums, far more digits for a space mission. An irrational number such as π or √2 can only ever be approximated by a fraction or a finite decimal.
Examples
22/7 is an approximation of π: π ≈ 22/7, but π ≠ 22/7.
√2 ≈ 1.414 is an approximation; 1.414 × 1.414 = 1.999396, not exactly 2.
Brahmagupta used √10 ≈ 3.162 as an approximation of π because it was easy to work with.
The Mesopotamians' approximation π ≈ 3.125 came from seeing that a circle is a little longer than the hexagon inside it.
Do not confuse: An approximation is not a mistake: it is a deliberate, close value. Rounding (3.14159 to 3.14) is one way to make an approximation.
Used in
measurement, values of π and square roots, estimation in everyday sums
A curved piece of a circle between two of its points.
Precisely: A connected part of a circle, fixed by two end points on it; of the two arcs they make, the smaller is the minor arc and the larger is the major arc.
How it works
Two points A and B split a circle into two arcs, one going each way round. The angle an arc makes at the centre tells you which is which: less than 180° for a minor arc, more than 180° for a major arc. The angle an arc makes at any point on the rest of the circle is half of its angle at the centre.
Examples
An arc that makes 70° at the centre makes 35° at any point on the rest of the circle.
Points A and B on a circle make a minor arc and a major arc; together the two arcs are the whole circle.
An arc of 180° at the centre is a semicircle, and the angle it makes at any point of the circle is 90°.
Between 12:00 and 12:15, the tip of the minute hand moves along an arc of 90° at the centre of the clock.
Do not confuse: An arc is curved and lies on the circle; a chord is straight and joins the same two end points across the inside.
Used in
angles in circles, arc length and sector area (Class 10), radian measure (Class 11)
How long a curved piece of a circle is: the fraction θ/360 of the whole circumference.
Precisely: The length of an arc that subtends θ° at the centre of a circle of radius r: l = 2πr × θ/360.
How it works
By the circle's rotational symmetry, equal angles at the centre cut off equal arcs. A semicircle (180°) is half the circumference, πr; a quarter circle (90°) is a quarter, πr/2; so an arc of θ° is θ/360 of 2πr.
Examples
In a circle of radius 6.3 m, an arc of 120° has arc length 2 × 22/7 × 6.3 × 120/360 = 13.2 m.
In a circle of radius 3.5 cm, an arc of 60° has arc length 22 × 1/6 = 11/3 ≈ 3.67 cm.
The arc length of a semicircle of radius r is πr, half of 2πr.
A sector of radius 14 cm and angle 75° has arc length about 18.33 cm, so its perimeter is 18.33 + 14 + 14 ≈ 46.33 cm.
Do not confuse: Arc length is a length along the curve; the chord joining the same two points is straight and always shorter.
Used in
running tracks and bends, the perimeter of sectors and petal shapes, radian measure (Class 11)
The amount of flat space a shape covers, counted in unit squares.
Precisely: The measure of a region of a plane, in square units; a 1 by 1 square has area 1 unit².
How it works
Count how many unit squares fit inside. A rectangle a by b holds ab of them. Other shapes are cut and rearranged into rectangles: a parallelogram gives base × height, a triangle half of that, and a circle, cut into thin slices, gives πr².
Examples
A rectangle 9 cm by 4 cm has an area of 36 cm², the same as a square of side 6 cm.
A triangle with base 12 cm and height 9 cm has area ½ × 12 × 9 = 54 cm².
A circle of radius 7 cm has area 22/7 × 7 × 7 = 154 cm².
A rhombus of side 3 can have area 9, 8.01 or 5.41: the sides alone do not fix the area of a 4-gon.
Do not confuse: Area is measured in square units (cm²), perimeter in plain units (cm). Doubling every side of a shape doubles its perimeter but makes its area four times as big.
Used in
land and floors, paint and tiles, Heron's and Brahmagupta's formulas, surface area and volume (Class 9 and 10)
OpenStax: Prealgebra 2e, 9.4 Use Properties of Rectangles, Triangles, and Trapezoids (area)
Argand plane
AR-gand playnClass 11
A flat grid for drawing complex numbers: a + ib is the point (a, b), with the real part along the across axis and the imaginary part up the other.
Precisely: The plane with a rectangular coordinate system in which each complex number a + ib is represented by the point (a, b); the x-axis is the real axis and the y-axis the imaginary axis.
How it works
Adding complex numbers moves the point like adding arrows; multiplying by i turns it a quarter turn round 0; the modulus is its distance from 0 and the conjugate is its reflection in the real axis.
Examples
On the Argand plane, 3 + 4i is the point (3, 4).
The number i sits one unit up the imaginary axis of the Argand plane.
Real numbers lie along the real axis of the Argand plane.
On the Argand plane, a number and its conjugate are reflections of each other in the real axis.
Do not confuse: The point (3, 4) on the Argand plane stands for the number 3 + 4i, not for the pair of real numbers 3 and 4 separately.
Used in
picturing complex numbers, the modulus and conjugate, rotations by multiplying by i
A sequence in which you add the same number each time to get the next term.
Precisely: A sequence a, a + d, a + 2d, ..., with first term a and common difference d; its nth term is tₙ = a + (n − 1)d.
How it works
Find d by subtracting any term from the next. The nth term is the first term plus (n − 1) steps of d. Plotted as points (n, tₙ), an AP always lies on a straight line.
Examples
The squares in the growing pattern, 1, 5, 9, 13, 17, ..., form an arithmetic progression with a = 1, d = 4 and tₙ = 4n − 3.
In the arithmetic progression 21, 18, 15, ..., −81 is the 35th term and 0 is the 8th.
A salary of ₹5,00,000 rising by ₹20,000 a year is an arithmetic progression that reaches ₹7,00,000 after 10 years.
The 2-digit multiples of 3, 12, 15, ..., 99, form an arithmetic progression of 30 terms with sum 1665.
If an arithmetic progression has 11th term 38 and 16th term 73, then d = 35/5 = 7 and the 31st term is 38 + 20 × 7 = 178.
Do not confuse: An AP adds a fixed number (d); a GP multiplies by a fixed number (r). 2, 4, 6, 8 is an AP; 2, 4, 8, 16 is a GP.
Used in
salaries, fares and savings plans, counting multiples, sums of APs (Class 10)
OpenStax: College Algebra 2e, 9.2 Arithmetic Sequences
assumed mean (and step deviation)
uh-SYOOMD meenClass 10
A guessed middle value that makes working out the mean easier; you correct the guess at the end.
Precisely: A value a (usually a central class mark) with deviations dᵢ = xᵢ − a; then x̄ = a + Σfᵢdᵢ ÷ Σfᵢ. With step deviations uᵢ = (xᵢ − a)/h, x̄ = a + h × Σfᵢuᵢ ÷ Σfᵢ.
How it works
Measuring from a middle value makes the numbers small (some negative), so the products are easy. Dividing by the class size h as well (step deviation) makes them smaller still. All three methods give the same mean.
Examples
With assumed mean a = 47.5 for the marks table, Σfᵢdᵢ = 435, so x̄ = 47.5 + 435/30 = 62.
With step deviations from the assumed mean 47.5 and h = 15, the uᵢ are −2, −1, 0, 1, 2, 3.
For 23, 25, 30 with assumed mean 25, the deviations are −2, 0 and 5.
Do not confuse: The assumed mean is only a starting point; it is almost never the answer, and a bad guess still gives the right mean, just with bigger numbers.
Used in
the mean of grouped data by hand, averaging from a base (a mental trick)
One of the two number lines of the coordinate plane: the x-axis goes across and the y-axis goes up and down.
Precisely: A reference line along which coordinates are measured; the x-axis and y-axis are perpendicular number lines that meet at the origin.
How it works
Each axis is a number line with 0 at the origin. The x-axis gives the first coordinate and the y-axis the second. A point on the x-axis has y = 0, and a point on the y-axis has x = 0.
Examples
The point (4.5, 0) lies on the x-axis, because it has not moved up or down at all.
The y-axis is the upright number line; every point on it looks like (0, y).
The two axes cut the plane into four quadrants.
On a graph of height against age, age runs along one axis and height along the other.
Do not confuse: A point on an axis belongs to no quadrant. And the axis is a whole line, not just the positive half drawn with an arrow.
Used in
the coordinate plane, graphs of equations, bar charts and line graphs