Algebraic Notation for ESAT Mathematics

Using letters and numbers in algebra

Algebraic terms are the building blocks of expressions and equations. They are created by combining numbers, letters, and brackets through multiplication or division. In mathematics, we use letters to represent variables or unknown values. To make these expressions easier to read and manipulate, we follow specific notation conventions that remove unnecessary symbols.

Multiplying in algebra

When multiplying variables or numbers together, the multiplication sign $\times$ is usually omitted. This prevents confusion with the letter $x$, which is commonly used as a variable.

1. Basic multiplication: The product $a \times b$ is written as $ab$ without any spaces or symbols between the letters.
2. Multiple variables: If three variables are multiplied, such as $p \times q \times r$, the result is written as $pqr$.
3. Variables and numbers: When a number is multiplied by a letter, the number always comes first. For example, $4 \times b = 4b$.
4. Repeated addition: It is important to distinguish between repeated addition and multiplication. Adding three identical variables together, $a + a + a$, is the same as $3 \times a$, which is written as $3a$.

When writing terms that contain both a number and several letters, the standard convention is to place the number (the coefficient) first, followed by the letters in alphabetical order. For example, $6 \times q \times r \times p$ should be written as $6pqr$.

Powers and indices

We use powers (or indices) to represent repeated multiplication of the same variable or number. This makes expressions much more compact.

1. Squares: Multiplying a variable by itself, $a \times a$, is written as $a^2$.
2. Cubes: Multiplying a variable by itself three times, $p \times p \times p$, is written as $p^3$.
3. Combined terms: If you have different variables being multiplied, some of which are repeated, you combine them. For instance, $p \times p \times q$ becomes $p^2q$.
4. Coefficients and powers: A term like $5 \times p \times p \times p$ is written as $5p^3$. Note that the power only applies to the letter $p$, not the number 5.

Dividing in algebra

In algebra, the division symbol $\div$ is rarely used. Instead, division is expressed using a fraction bar. The expression $a \div b$ is written as $\frac{a}{b}$. This notation is much clearer when dealing with complex algebraic fractions and helps in identifying factors that can be cancelled out.

Using brackets

Brackets are used to group terms together. When a power is applied to a bracketed term, it applies to everything inside the bracket. For example, $(ab)^2$ means $(ab) \times (ab)$. When expanded, this results in $a \times b \times a \times b$, which simplifies to $a^2 \times b^2$, or $a^2b^2$.