MATH DICTIONARY Z

Zero :

Originally, one of the integers in the real number system that is smaller than 1 but greater than -1. The only real number that is neither positive nor negative and also it indicates no quantity, size, or magnitude.

Zero Property of Addition :

The property that states that the sum of a number and 0 is that same number. 

k + 0 = k

The above property is true for all real values of k.

This property is sometimes called the identity property of addition.

Zero Property of Multiplication :

The property that states that the product of any number and 0 is always 0

k × 0 = 0

The above property is true for all real values of k.

Zero Index :

Any non zero number to the power of zero is equal to one.

Example 1 :

5⁰ = 1

8⁰ = 1

Zero - Importance :

The importance of the number zero goes beyond the real numbers because it serves as the additive identity also for the complex numbers: If z is any complex number, then

z + 0 = z

Also, zero could be considered as the ”zero function” that for all values of the variable x equals to zero. In this interpretation zero is also the additive identity in the set of all functions.

Zero Factor Property :

Also called zero product property. If for two numbers (real or complex) a and b,

a · b = 0,

then either a = 0 or b = 0 or both are zero.

This property of numbers is one of the most important tools in solving algebraic or trigonometric equations.

Example 2 :

x³ + 3x² - 4x - 12 = 0

Factoring the polynomial on the left side by grouping, we get the equation

(x - 2)(x + 2)(x + 3) = 0

which by the zero factor property results in three linear equations

x - 2 = 0, x + 2 = 0, x + 3 = 0

with the solutions

x = 2, x = -2, x = -3

Example 3 :

sin2x - cosx = 0

Using the double angle formula for the sine function and factoring we get cos

x(2sinx - 1) = 0

and again, by the zero factor property this results in two simpler equations

cosx = 0

sinx = 1/2

The solutions on the base interval [0, 2π) are :

x = π/2, 3π/2, π/6, 5π/6

The zero factor property is not true for other mathematical object, for example for matrices : If the product A·B of two matrices is the zero matrix then none of the matrices A or B are necessarily the zero matrices themselves.

Zero Matrix :

A matrix of arbitrary size mxn such that all of the entries are zeros. Example :

This matrix is the additive identity in the set of all matrices of the same size: Adding this matrix to any other matrix does not change it.

Zeros of the Equation :

Also called roots or solutions of the equation. Any real (or complex) number that substituted into the equation results in numeric identity. The number x = 2 is the zero of the equation

x² - x - 2 = 0

because substituting that value into the left side of the equation results in numeric identity

0 = 0

Every polynomial equation of degree n ≥ 1 has exactly n zeros, if we accept complex zeros and count them according to multiplicity. See also Fundamental Theorem of Algebra.

Zero Subspace :

The subspace of a vector space that consists of a single zero vector x = 0. Zero subspace is the part of any vector space.

Zero Transformation :

The (linear) transformation between any vector spaces V and W such that T(u) = 0 for any vector u ∈ V.

Zero Vector :

The vector that has all zero components :

0 = (0, 0, · · · , 0)

This vector plays the role of the additive unity for any vector space :

x + 0 = 0 + x = x

for any vector x of the vector space.

Z-Axis :

In three-dimensional Cartesian coordinate system, in addition to x- and y-axes, there is also the z-axis, that is perpendicular to the plane formed by these two axes.

Z-Coordinate :

Any point in the space has three orthogonal projections, corresponding to three axes. The projection onto the z-axis gives the z-coordinate of the point that is always written on the third position : For the point (1, 2 , 3) the third entry 3 is the z-coordinate.

Z-Score :

Suppose x is a random variable and assume x̄ is its mean and s its standard deviation. The quantity

z = (x - x̄)/s

is the z-score (or standardized value) of the variable x. If the variable x belongs to a normal distribution with the mean x̄ and standard deviation s, then the variable z (the z-score) belongs to standard normal distribution: its mean is zero and standard deviation is 1. The z-score measures the distance of the value x from the mean in terms of standard deviation units. z-scores are also sometimes called z statistic.

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