Understanding and performing calculations with values in standard form is a fundamental skill in maths, particularly relevant for CIE IGCSE students. Standard form, or scientific notation, is a concise way of expressing very large or very small numbers, using the format $A \times 10^n$, where $n$ is an integer and 1 \leq A < 10. This section delves into performing calculations with numbers in standard form, illustrating the process through examples and providing a clear explanation of applying this concept in various contexts.

**Understanding Standard Form**

Standard form is used to simplify the representation and calculation of very large or very small numbers. It is structured as $A \times 10^n$, where:

- $A$ is a number between 1 and 10 (inclusive of 1 and exclusive of 10).
- $n$ is an integer that signifies the power to which 10 is raised.

Calculations involving numbers in standard form can include addition, subtraction, multiplication, and division. The rules for each operation take into account the power of 10 and the coefficient $A$.

**Multiplication**

When multiplying numbers in standard form, you multiply the coefficients ($A$ values) and add the exponents ($n$ values) of the $10^n$ term.

**Example:**

Multiply $2.5 \times 10^4)$ by $3.2 \times 10^3$.

**Solution:**

**Division**

Division involves dividing the coefficients and subtracting the exponent of the divisor from the exponent of the dividend.

**Example:**

Divide $6.4 \times 10^6$ by $2 \times 10^2$.

**Solution:**

**Addition and Subtraction**

For addition and subtraction, the exponents must be the same. If they are not, adjust the numbers so that they have the same exponent before performing the operation on the coefficients.

**Example:**

Add $3 \times 10^4$ to $4.5 \times 10^3$.

First, express $4.5 \times 10^3$ as $0.45 \times 10^4$.

Then, $3 \times 10^4 + 0.45 \times 10^4 = 3.45 \times 10^4$.

**Worked Examples**

**1. Multiplication in Standard Form**

**Question:** Multiply $5.6 \times 10^3$ by (2.5 \times 10^2).

**Solution:**

$(5.6 \times 2.5) \times 10^{3+2}$

$14 \times 10^5$

$= 1.4 \times 10^6$

**2. Division in Standard Form**

**Question:** Divide $9 \times 10^5$ by $3 \times 10^2$.

**Solution:**

$9 \div 3 \times 10^{5-2}$

$3 \times 10^3$

**3. Addition in Standard Form**

**Question:** Add $2.3 \times 10^4$ and $3.7 \times 10^3$.

**Solution:**

First, convert $3.7 \times 10^3$ to $0.37 \times 10^4$.

$(2.3 + 0.37) \times 10^4$

$2.67 \times 10^4$

$= 2.67 \times 10^4$

**4. Subtraction in Standard Form**

**Question:** Subtract $1.2 \times 10^4$ from $6.5 \times 10^4$.

**Solution:**

$(6.5 - 1.2) \times 10^4$

$5.3 \times 10^4$

$= 5.3 \times 10^4$

**Applying Standard Form in Different Contexts**

Standard form is not just a mathematical trick; it has practical applications in science, engineering, and finance, among other fields, to handle very large numbers, such as the distance between stars, or very small numbers, like the mass of a virus.

**Scientific Applications**

In physics, distances within the universe can be vast. The distance from Earth to the nearest star, Proxima Centauri, is approximately $4.24 \times 10^{16}$ meters. Expressing this distance in standard form makes it comprehensible and manageable.

**Engineering Applications**

Engineers use standard form to deal with quantities ranging from the very large, such as the wattage of a power station, to the very small, such as the resistance of microcircuits.

**Financial Applications**

In economics and finance, standard form helps in the calculation of national budgets and the GDP of countries, which can run into the trillions, as well as in calculating interest rates on loans and savings.