Basic Conventions of Algebra
Algebra uses letters and symbols to represent numbers and relationships. Understanding the basic conventions of algebra helps expressions and equations to be written clearly and interpreted correctly.
Letters are used to represent variables, which are numbers that can change or are unknown. For example, \( x \) may represent any number. When a number is written next to a letter, it means multiplication. So \( 3x \) means \( 3 \times x \).
Multiplication is often written without a symbol. For example, \( ab \) means \( a \times b \), and \( 2(a + 5) \) means \( 2 \times (a + 5) \). This convention helps keep algebraic expressions compact and easy to read.
Division can be written using a fraction. For example:
$$
\frac{x}{4}
$$
This means \( x \) divided by \( 4 \). Writing division as a fraction avoids confusion, especially in more complex expressions.
Powers are written using indices. For example, \( x^2 \) means \( x \times x \), and \( x^3 \) means \( x \times x \times x \). The index applies only to the letter or expression directly before it unless brackets are used.
Brackets show the order in which operations should be carried out. For example, \( 2x + 3 \) is different from \( 2(x + 3) \). In the second case, everything inside the brackets is multiplied by \( 2 \).
Like terms are terms that have the same variable part. For example, \( 3x \) and \( 5x \) are like terms and can be combined:
$$
3x + 5x = 8x
$$
However, \( x \) and \( x^2 \) are not like terms and cannot be combined.
An equation shows that two expressions are equal. For example:
$$
2x + 1 = 9
$$
An equation can be solved to find the value of the variable.
Following these algebraic conventions ensures expressions are written unambiguously and makes calculations, simplification and problem solving much clearer.