One of the eight parts into which the three coordinate planes cut space, like the four quadrants of a flat grid but in three dimensions.
Precisely: Each of the eight regions into which the xy-, yz- and zx-planes divide space, named by the signs of the coordinates of its points; the first octant has x, y and z all positive.
How it works
Each coordinate can be positive or negative, so there are 2 × 2 × 2 = 8 sign patterns, one for each octant. The signs of a point's coordinates tell you which octant it is in.
Examples
The point (2, 3, 4) lies in the first octant.
The point (−1, 2, −3) lies in the octant where x and z are negative and y is positive.
The corner of a room is like the origin, with the room itself filling one octant.
A point with a coordinate of 0 lies on a coordinate plane, in no octant.
Do not confuse: An octant is an eighth of space; a quadrant is a quarter of a flat plane.
Used in
three-dimensional geometry, locating points in space
Matching the things in one group with the things in another, one with one, so that nothing is left over on either side.
Precisely: A pairing between two collections in which each member of the first is matched with exactly one member of the second, and each member of the second with exactly one of the first.
How it works
If the matching uses up both groups exactly, they have the same number of things, even if you cannot count. This is how counting began, long before number words.
Examples
A herder dropped one pebble in a pot for each cow that left: a one-to-one correspondence between pebbles and cows.
If every chair in the room has exactly one student and no one is standing, there is a one-to-one correspondence between chairs and students.
Counting "1, 2, 3, 4" while touching four apples sets up a one-to-one correspondence between the apples and the numbers 1 to 4.
Do not confuse: Two groups can have a one-to-one correspondence even if the things are different kinds; it compares how many, not what they are.
Used in
counting and natural numbers, comparing sizes of groups, functions (Class 11 and 12)
In a right triangle, from one of the sharp angles: the opposite side faces the angle, and the adjacent side is next to it (not the hypotenuse).
Precisely: For an acute angle A of a right triangle, the opposite side is the side across from A, the adjacent side is the other side that forms A, and the hypotenuse is the longest side, opposite the right angle.
How it works
Stand at the angle: the side you touch that is not the hypotenuse is adjacent; the side you look across at is opposite. Which side is which changes when you move to the other acute angle.
Examples
In △ABC right-angled at B, for angle A the opposite side is BC and the adjacent side is AB.
For angle C of the same triangle the roles swap: the opposite side is AB and the adjacent side is BC.
Looking up at a tree, the tree is the opposite side of your angle of elevation and the ground to its foot is the adjacent side.
Do not confuse: The hypotenuse is never called adjacent, even though it touches the angle; and the sides are named from one angle, not fixed.
OpenStax: College Algebra 2e, 2.1 Key Terms (origin)
outcome
OWT-kumClass 9
One possible result of a random experiment.
Precisely: A single element of the sample space of a random experiment.
How it works
List every outcome once, with none missing and none repeated; together they make the sample space. An event is a group of outcomes, and the favourable outcomes are the ones in the event.
Examples
Rolling a die has six outcomes: 1, 2, 3, 4, 5 and 6.
Tossing two coins has four outcomes: HH, HT, TH and TT.
Choosing a snack (samosa, pakora or bhaji) and a drink (chai or lassi) has 3 × 2 = 6 outcomes.
A hockey match has three outcomes for a team: win, lose or draw.
Do not confuse: HT and TH are different outcomes when two coins are tossed (first coin heads or first coin tails), even though both have one head.