Basis : The basis of a vector space is a set of linearly independent vectors, so that every vector of the vector space can be written as a linear combination of the vectors in the basis. For example $\{\left(\begin{smallmatrix}1\\0\\0\\\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\1\\0\\\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\0\\1\\\end{smallmatrix}\right)\}$ is a basis of $\mathbb{R}^3.$ In a prove by induction basis refers to the first step in which you have to prove that the result is true for a particular number $n$ (usually $n=0$ or $n=1$.)

Division : Division is the mathematical operation that divides objects equally into groups. More generally $x\div y$ is defined as the number that if multiplied by $y$ equals $x.$

Exponent : In a power $a^x$ the number $x$ is called the exponent of the power.

Fractional exponent : The definition of a power with a fractional exponent is $x^\frac{a}{b}:=\sqrt[b]{{x}^a}.$ For example $8^{\frac{2}{3}}=\sqrt[3]{{8}^2}=4.$

Half : A Half is the number equal to $\frac{1}{2}=0.5.$

Power : A power is a number of the form $a^b.$ $b$ is called the exponent of the power and $a^b$ is called a power of $a$. For natural numbers $b$ the number $a^b$ is an abbreviation for successively multiplying $a$ by itself $b$ times. For example $2^3=2\cdot 2\cdot 2=8.$ For fractional exponents $b=\frac{p}{q}$ the number $a^{\frac{p}{q}}$ is defined as $\sqrt[q]{a^p}.$ For arbitrary real exponents $b$ the power $a^b$ is defined as the limit of $a^{b_n}$ with rational $b_n$ that converge towards $b.$

Subtraction : Subtraction is the mathematical operation that describes decreasing a number by an amount equal to a second number. The mathematical symbol for subtraction is the minus sign $-.$ The term subtraction is also used for a generalization of this basic operation on numbers to functions, vectors and matrices.

Number : A number $x$ is a mathematical symbol representing a quantity.

Square : A square is a quadrilateral with four right angles and four equal length sides. A square is a special case of a rectangle and a special case of a rhombus.

Square root : The square root of $x$ denoted by $\sqrt{x}$ is the positive number such that $(\sqrt{x})^2=\sqrt{x}\cdot \sqrt{x}=x.$ For example $\sqrt{9}=3.$

Basis : The basis of a vector space is a set of linearly independent vectors, so that every vector of the vector space can be written as a linear combination of the vectors in the basis. For example $\{\left(\begin{smallmatrix}1\\0\\0\\\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\1\\0\\\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\0\\1\\\end{smallmatrix}\right)\}$ is a basis of $\mathbb{R}^3.$ In a prove by induction basis refers to the first step in which you have to prove that the result is true for a particular number $n$ (usually $n=0$ or $n=1$.)

Division : Division is the mathematical operation that divides objects equally into groups. More generally $x\div y$ is defined as the number that if multiplied by $y$ equals $x.$

Exponent : In a power $a^x$ the number $x$ is called the exponent of the power.

Power : A power is a number of the form $a^b.$ $b$ is called the exponent of the power and $a^b$ is called a power of $a$. For natural numbers $b$ the number $a^b$ is an abbreviation for successively multiplying $a$ by itself $b$ times. For example $2^3=2\cdot 2\cdot 2=8.$ For fractional exponents $b=\frac{p}{q}$ the number $a^{\frac{p}{q}}$ is defined as $\sqrt[q]{a^p}.$ For arbitrary real exponents $b$ the power $a^b$ is defined as the limit of $a^{b_n}$ with rational $b_n$ that converge towards $b.$