Index Number definition
An index number is the small number written above and to the right of a base number or algebraic term. It tells you how many times the base is multiplied by itself. Index numbers are also called powers or exponents. For example, in 5 squared, the base is 5 and the index number is 2. This means 5 times 5, which equals 25.
Index notation is a compact way of writing repeated multiplication. Instead of writing 3 times 3 times 3 times 3, you can write 3 to the power of 4. The index number 4 tells you that the base 3 appears as a factor four times. This notation is useful because repeated multiplication can become long and difficult to read, especially in algebra or science.
Index numbers are used in algebra, number, standard form, surds, graphs, sequences, compound growth, exponentials, logarithms, calculus, science and real-life modelling. At GCSE level, students use index numbers when working with squares, cubes, powers of 10, standard form and the laws of indices. At A-Level, indices connect strongly to exponential functions, logarithms, differentiation and integration.
Some index numbers have special names. A power of 2 is called a square, such as 7 squared. A power of 3 is called a cube, such as 4 cubed. This language connects indices to geometry. A square number can represent the area of a square, and a cube number can represent the volume of a cube. For example, 4 cubed equals 4 times 4 times 4, which equals 64.
Index laws are rules for working with powers. If the bases are the same, multiplying powers means adding the indices. For example, x to the power of 3 times x to the power of 2 equals x to the power of 5. Dividing powers with the same base means subtracting the indices. For example, x to the power of 5 divided by x squared equals x cubed. These rules help simplify algebraic expressions.
The zero index and negative indices are also important. Any non-zero number to the power of 0 equals 1. For example, 8 to the power of 0 equals 1. A negative index means a reciprocal. For example, 2 to the power of -3 equals 1 over 2 cubed, which is 1 over 8. These ideas can feel unusual at first, but they follow from the same index patterns used with division.
Index numbers are central to standard form. Very large and very small numbers can be written using powers of 10. For example, 4,500,000 can be written as 4.5 times 10 to the power of 6. This makes the number easier to read, compare and calculate with. Standard form is used in science, engineering, astronomy, finance and many calculator-based problems.
Related ideas include powers, exponents, bases, squares, cubes, index laws, standard form, surds, exponential functions and logarithms. A clear definition of index number helps students understand that an index is not decoration. It changes the value of the expression and gives a powerful way to describe repeated multiplication, growth and scale.