Standard Form

Some numbers are awkward to write out in full. The distance to the Sun is about $150\,000\,000$ km; the width of a human hair is about $0.000\,07$ m. Counting all those zeros is slow and easy to get wrong.

Standard form is a tidy shorthand. Every number is written as $a \times 10^n,$ where $a$ is at least 1 and less than 10, and $n$ is an integer.

So $150\,000\,000$ becomes $1.5 \times 10^8$, and $0.000\,07$ becomes $7 \times 10^{-5}$. The power of 10 carries all the size information, while $a$ holds just the meaningful digits.

The rule on $a$ is strict: $3.6 \times 10^4$ is correct, but $36 \times 10^3$ is not in standard form — $36$ is not less than 10. Getting $a$ into range is the key habit of this whole topic.

For a large number — one bigger than 10 — the power of 10 is positive.

The method: place a decimal point so that exactly one non-zero digit sits in front of it, then count how many places the point has moved. That count is the power.

For $4\,800\,000$: place the point after the 4 to get $4.8$. The point has moved 6 places, so the answer is $4.8 \times 10^6$.

A quick sense-check: a large number always has a positive power, and the power is one less than the number of digits before the point in the original. $4\,800\,000$ has 7 digits, and the power is 6 — that agrees.

For a small number — one between 0 and 1 — the power of 10 is negative. This is where the negative indices you have met come into their own.

The method is the mirror image. Move the decimal point right until one non-zero digit sits in front of it, count the places, and make the power negative.

For $0.000\,36$: move the point right until it sits after the 3, giving $3.6$. The point moved 4 places, so the answer is $3.6 \times 10^{-4}$.

A negative power does not make the number negative — $3.6 \times 10^{-4}$ is a small positive number, $0.000\,36$. The minus sign in the power simply signals "this is a number less than 1".

Standard form makes ordering numbers wonderfully quick — far quicker than counting zeros.

The method has two steps. First, compare the powers of 10 — the bigger power means the bigger number. Only if two numbers share the same power do you look at the number part $a$ to break the tie.

So $8 \times 10^3$, $2 \times 10^5$ and $9 \times 10^5$ in order are: $8 \times 10^3$ first (smallest power), then $2 \times 10^5$ and $9 \times 10^5$ — same power, so 2 comes before 9.

Take care with negative powers. $10^{-2}$ is bigger than $10^{-5}$, just as $-2$ is bigger than $-5$ on a number line.

One of the best things about standard form is how neatly it multiplies and divides — you deal with the number parts and the powers separately.

• To multiply: multiply the $a$ values, and add the powers (the multiplying index law).
• To divide: divide the $a$ values, and subtract the powers.

So $(3 \times 10^4) \times (2 \times 10^5) = (3 \times 2) \times 10^{4+5} = 6 \times 10^9$.

Sometimes the number parts give a result outside the 1-to-10 range. $(6 \times 10^3) \times (5 \times 10^2) = 30 \times 10^5$ — but $30$ is too big, so tidy it: $30 = 3 \times 10^1$, giving $3 \times 10^6$. Always finish by checking $a$ is in range.

Standard form is used far beyond the maths classroom:

• Science writes almost every measurement in standard form — the mass of an atom, the speed of light, the distance to a star.
• Calculators display very large or very small results in standard form, so reading it fluently is a real-world skill.
• Algebra in later stages uses the same index laws on powers of 10 that you have practised here.
• Computing and data describe storage and processing speeds with the same shorthand, such as $5 \times 10^9$ operations.

If standard form feels shaky later, it is usually the rule on $a$, or an index law for the powers, that needs a refresh. The "tidy $a$ into range" habit is the one to lock in.