Writing numbers in scientific notation (standard form), converting between ordinary and scientific notation, and multiplying and dividing numbers written in scientific notation.

What this dot point is asking

The SQA wants you to write a very large or very small number in scientific notation (also called standard form), convert a number in standard form back to an ordinary number, and multiply or divide numbers written in scientific notation, giving the answer correctly in that form.

What scientific notation is

Scientific notation writes any number as

where $1 \le a < 10$ and $n$ is an integer. The front number $a$ is sometimes called the mantissa. The form makes very large and very small numbers easy to write and compare: $5\,300\,000 = 5.3 \times 10^6$ and $0.0000072 = 7.2 \times 10^{-6}$.

The sign of the power tells you the size. A positive power is a large number, and the power counts how many places the decimal point moves to the right of the front digit. A negative power is a small number (between $0$ and $1$), and the power counts how many places the point moves to the left.

Converting back to an ordinary number

To undo scientific notation, move the decimal point the number of places given by the power: right for a positive power, left for a negative power, filling with zeros.

Multiplying and dividing in scientific notation

Because $10^m \times 10^n = 10^{m+n}$ and $10^m \div 10^n = 10^{m-n}$, you handle the front numbers and the powers of ten separately.

The adjustment step matters: if the front number comes out as $24$, you rewrite it as $2.4 \times 10^1$ and absorb that extra power of ten. If it comes out as $0.5$, you rewrite it as $5 \times 10^{-1}$.

When the front numbers do not divide neatly, the same procedure still applies, and you adjust the result at the end. For instance, $\dfrac{3 \times 10^4}{6 \times 10^9} = 0.5 \times 10^{4 - 9} = 0.5 \times 10^{-5}$, and because $0.5$ is less than $1$ you rewrite it as $5 \times 10^{-1}$, giving $5 \times 10^{-1} \times 10^{-5} = 5 \times 10^{-6}$.

Comparing numbers in scientific notation

Standard form makes it quick to put numbers in order of size. First compare the powers of ten: the larger the power, the larger the number. Only when two numbers have the same power do you need to look at the front numbers. So $7 \times 10^5$ is larger than $9 \times 10^4$, because $10^5$ beats $10^4$ even though $9$ is bigger than $7$. For very small numbers a less negative power is larger, so $3 \times 10^{-4}$ is larger than $8 \times 10^{-6}$.

Examples in context

Scientific notation is the working language of science and engineering. The distance from the Earth to the Sun is about $1.5 \times 10^{11}$ m, and the diameter of a red blood cell is about $8 \times 10^{-6}$ m. Astronomers multiply such numbers constantly: light travelling at $3 \times 10^8$ m/s for $5 \times 10^2$ s covers $(3 \times 10^8)(5 \times 10^2) = 15 \times 10^{10} = 1.5 \times 10^{11}$ m, which is roughly the Earth-Sun distance. Working in standard form keeps the arithmetic manageable.

Try this

Q1. Write $72\,000$ in scientific notation. [1 mark]

• Cue. $7.2 \times 10^4$.

Q2. Write $2.5 \times 10^{-3}$ as an ordinary number. [1 mark]

• Cue. $0.0025$.

Q3. Calculate $(4 \times 10^6) \times (5 \times 10^2)$ in scientific notation. [2 marks]

• Cue. $20 \times 10^8 = 2 \times 10^9$.