Place value, ordering integers and decimals, multiplying and dividing by powers of ten, and writing and calculating with numbers in standard form.

What this dot point is asking

AQA wants you to understand place value, order integers and decimals, multiply and divide by powers of ten, and write and calculate with numbers in standard form. Standard form is how very large and very small numbers (populations, atomic sizes, distances in space) are handled compactly, and it is tested on both papers, often with the index laws.

Place value and ordering

Each digit's value depends on its column: in $3\,407$ the $3$ is three thousands, the $4$ is four hundreds. After the decimal point come tenths, hundredths and thousandths. To order decimals, compare digits from the left, column by column: $0.3$ is larger than $0.27$ because the tenths digit $3$ beats $2$, even though $27$ looks bigger than $3$. A common technique is to give every number the same number of decimal places ($0.30$ versus $0.27$) before comparing.

Multiplying and dividing by powers of ten

Multiplying by $10$ moves every digit one place to the left, so $4.6 \times 10 = 46$; multiplying by $100$ moves two places, and by $1000$ three places. Dividing reverses this: $4.6 \div 100 = 0.046$. Think of the digits moving rather than "adding zeros", which fails for decimals. So $0.072 \times 1000 = 72$ (digits move three places left) and $580 \div 10\,000 = 0.058$.

Standard form

To convert $52\,000$ to standard form, place the decimal after the first digit to get $5.2$, then count how many places it moved: five places, and the number is large, so $52\,000 = 5.2 \times 10^4$. For a small number like $0.0008$, the first non-zero digit is $8$, the point moves four places, and the number is small, so $0.0008 = 8 \times 10^{-4}$.

Calculating in standard form

For addition and subtraction, it is usually easiest to convert both numbers to ordinary form (or the same power of ten) first, combine, then return to standard form. For multiplication and division, multiply or divide the number parts and add or subtract the powers, then adjust if the number part falls outside the range $1$ to $10$.

Adding and subtracting in standard form

To add $3 \times 10^4$ and $5 \times 10^3$, rewrite both with the same power: $3 \times 10^4 = 30 \times 10^3$, so the sum is $30 \times 10^3 + 5 \times 10^3 = 35 \times 10^3 = 3.5 \times 10^4$. The key step is matching the powers of ten before combining the number parts; you cannot simply add the number parts when the powers differ. The final answer must always be tidied back into proper standard form, with the first part between $1$ and $10$.

Why standard form matters

Standard form is the language of science and large-data contexts: the mass of an electron is about $9.1 \times 10^{-31}\,\text{kg}$ and the distance to the Sun is about $1.5 \times 10^{11}\,\text{m}$. Writing these as ordinary decimals would be error-prone and hard to compare. Standard form also makes the size of a number instantly readable from its power of ten, so $10^{11}$ is clearly far larger than $10^{8}$, and it lets you multiply and divide huge or tiny quantities using the simple index laws rather than counting zeros.