Real numbers and scientific notation

What you will learn

classify numbers as natural, integer, rational, or irrational and explain the hierarchy,
represent rational and irrational numbers on the real number line,
convert between ordinary notation and scientific notation,
perform multiplication and division with numbers in scientific notation,
apply scientific notation to real-world contexts involving very large or very small quantities.

Worked example 0 Real-world example: how far is a light-year?

Light travels at approximately $3 \times 10^8$ metres per second. A year has about $3.15 \times 10^7$ seconds.

Distance = speed $\times$ time = $(3 \times 10^8) \times (3.15 \times 10^7)$ .
Multiply the decimal parts: $3 \times 3.15 = 9.45$ .
Add the exponents: $10^8 \times 10^7 = 10^{15}$ .
Combine: one light-year $\approx 9.45 \times 10^{15}$ m.

Key idea: scientific notation lets us multiply enormous quantities by working with small decimals and adding exponents — no need to write out fifteen zeros.

1. The real number system

Every number you have met so far fits into a hierarchy:

\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

$\mathbb{N}$ — natural numbers: $0, 1, 2, 3, \ldots$
$\mathbb{Z}$ — integers: $\ldots, -2, -1, 0, 1, 2, \ldots$
$\mathbb{Q}$ — rational numbers: any number that can be written as $\dfrac{a}{b}$ with $a, b \in \mathbb{Z}$ and $b \neq 0$ . Their decimals either terminate or recur.
$\mathbb{R}$ — real numbers: all of $\mathbb{Q}$ together with the irrational numbers (decimals that neither terminate nor recur).

The real number line, showing where rational and irrational numbers sit.

Worked example 1 Classifying numbers

Classify each number as rational or irrational: $\;\dfrac{7}{11},\; \sqrt{36},\; \pi + 1,\; 0.\overline{142857},\; \sqrt{10}$ .

$\dfrac{7}{11}$ — ratio of integers, so rational.
$\sqrt{36} = 6$ — a whole number, so rational.
$\pi + 1$ — since $\pi$ is irrational, adding $1$ (rational) still gives an irrational number.
$0.\overline{142857}$ — repeating decimal, so rational (in fact it equals $\dfrac{1}{7}$ ).
$\sqrt{10}$ — $10$ is not a perfect square, so $\sqrt{10}$ is irrational.

2. Scientific notation

Scientific notation writes a number as:

Scientific notation form

a \times 10^n, \qquad 1 \leq a < 10, \quad n \in \mathbb{Z}.

The coefficient $a$ has exactly one non-zero digit before the decimal point. The exponent $n$ is a positive integer for large numbers and a negative integer for small numbers.

Worked example 2 Converting to scientific notation

Write $47\,200\,000$ in scientific notation.

Place the decimal after the first non-zero digit: $4.72$ .
Count how many places the decimal moved: $7$ places to the left.
The exponent is positive because the original number is large: $n = 7$ .

$47\,200\,000 = 4.72 \times 10^7.$

Worked example 3 Small numbers in scientific notation

Write $0.000\,003\,6$ in scientific notation.

Place the decimal after the first non-zero digit: $3.6$ .
Count how many places the decimal moved: $6$ places to the right.
The exponent is negative because the original number is small: $n = -6$ .

$0.000\,003\,6 = 3.6 \times 10^{-6}.$

Worked example 4 Converting from scientific notation to ordinary form

Write $8.05 \times 10^{-4}$ in ordinary notation.

The exponent is $-4$ , so move the decimal $4$ places to the left:

$8.05 \times 10^{-4} = 0.000\,805.$

3. Operations with scientific notation

When multiplying or dividing numbers in scientific notation, handle the coefficients and powers of ten separately.

Operations in scientific notation

Multiplication

(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}.

Adjust if $a \times b \geq 10$ .

Division

\dfrac{a \times 10^m}{b \times 10^n} = \dfrac{a}{b} \times 10^{m-n}.

Adjust if $\dfrac{a}{b} < 1$ .

Worked example 5 Multiplying in scientific notation

Calculate $(4.5 \times 10^6) \times (2 \times 10^3)$ .

Multiply coefficients: $4.5 \times 2 = 9$ .
Add exponents: $10^6 \times 10^3 = 10^9$ .
Result: $9 \times 10^9$ .

Since $1 \leq 9 < 10$ , no adjustment needed.

Worked example 6 Multiplying with adjustment

Calculate $(6 \times 10^4) \times (5 \times 10^7)$ .

Multiply coefficients: $6 \times 5 = 30$ .
Add exponents: $10^4 \times 10^7 = 10^{11}$ .
Preliminary result: $30 \times 10^{11}$ .
Adjust: $30 = 3.0 \times 10^1$ , so the answer is $3.0 \times 10^{12}$ .

Worked example 7 Dividing in scientific notation

Calculate $\dfrac{8.4 \times 10^9}{2.1 \times 10^3}$ .

Divide coefficients: $8.4 \div 2.1 = 4$ .
Subtract exponents: $10^{9-3} = 10^6$ .
Result: $4 \times 10^6$ .

Practice

Fluency

Tier 1: basic skills

Classify as rational or irrational: $\dfrac{5}{8}$ .
Classify: $\sqrt{50}$ .
Classify: $-\sqrt{81}$ .
Classify: $0.\overline{36}$ .
Write $93\,000\,000$ in scientific notation.
Write $0.000\,072$ in scientific notation.
Write $5.03 \times 10^5$ in ordinary notation.
Write $1.7 \times 10^{-3}$ in ordinary notation.
Calculate $(3 \times 10^4) \times (2 \times 10^5)$ . Give your answer in scientific notation.
Calculate $\dfrac{9.6 \times 10^8}{3.2 \times 10^2}$ . Give your answer in scientific notation.

Reasoning

Tier 2: mixed practice

Place $\sqrt{5}$ , $\dfrac{7}{3}$ , and $\pi$ on a number line. Which is the largest?
Show that $\sqrt{2} + \sqrt{2}$ is irrational.
The mass of the Earth is approximately $5.97 \times 10^{24}$ kg and the mass of the Moon is approximately $7.35 \times 10^{22}$ kg. How many times heavier is the Earth than the Moon? Give your answer to the nearest whole number.
A human hair is about $7 \times 10^{-5}$ m wide. Express this in micrometres ( $1\;\mu\text{m} = 10^{-6}$ m).
Calculate $(7.2 \times 10^{-3}) \times (4 \times 10^5)$ and give the result in scientific notation.
Explain why the sum of a rational number and an irrational number is always irrational.
Between which two consecutive tenths does $\sqrt{18}$ lie? Use squaring to justify.
The distance from the Sun to Neptune is $4.5 \times 10^{12}$ m. Light travels at $3 \times 10^8$ m/s. How many seconds does sunlight take to reach Neptune?

Reasoning

Tier 3: explain and apply

Is $\sqrt{2} \times \sqrt{8}$ rational or irrational? Justify your answer.
A nanotechnology lab measures objects on the scale of $10^{-9}$ m. Express $4.2 \times 10^{-7}$ m in terms of nanometres.
Find two irrational numbers whose product is rational. Explain why this does not contradict the definition of irrational numbers.
The Australian national debt is approximately $9 \times 10^11 $. If the population is$ 2.6 \times 10^7$, estimate the debt per person. Give your answer in ordinary notation to the nearest dollar.
Explain why $0.\overline{9} = 1$ and what this tells us about the boundary between rational and irrational numbers.

Challenge

Reasoning

Harder reasoning

Prove that if $r$ is rational and $x$ is irrational, then $r + x$ is irrational. (Hint: assume the opposite and derive a contradiction.)
The observable universe has a radius of approximately $4.4 \times 10^{26}$ m. Estimate its volume in cubic metres using $V = \dfrac{4}{3}\pi r^3$ , and express your answer in scientific notation to two significant figures.
Simplify $\dfrac{(2.4 \times 10^5)^2}{6 \times 10^3}$ and give the answer in scientific notation.
A computer performs $3.6 \times 10^{12}$ operations per second. How many operations can it perform in one year ( $3.15 \times 10^7$ seconds)? If each operation processes $8 \times 10^{-9}$ seconds of audio, how many hours of audio can the computer process per year?

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Tier 1

Rational. $\dfrac{5}{8} = 0.625$ , a terminating decimal.
Irrational. $50$ is not a perfect square.
Rational. $\sqrt{81} = 9$ , so $-\sqrt{81} = -9$ , an integer.
Rational. Repeating decimal; $0.\overline{36} = \dfrac{4}{11}$ .
$9.3 \times 10^7$ .
$7.2 \times 10^{-5}$ .
$503\,000$ .
$0.0017$ .
$6 \times 10^9$ . Method: $3 \times 2 = 6$ ; $10^4 \times 10^5 = 10^9$ .
$3 \times 10^6$ . Method: $9.6 \div 3.2 = 3$ ; $10^{8-2} = 10^6$ .

Tier 2

$\sqrt{5} \approx 2.24$ , $\dfrac{7}{3} \approx 2.33$ , $\pi \approx 3.14$ . Largest is $\pi$ .
$\sqrt{2} + \sqrt{2} = 2\sqrt{2}$ . If this were rational, then $\sqrt{2} = \dfrac{r}{2}$ for some rational $r$ , making $\sqrt{2}$ rational — contradiction. So $2\sqrt{2}$ is irrational.
$\dfrac{5.97 \times 10^{24}}{7.35 \times 10^{22}} = \dfrac{5.97}{7.35} \times 10^2 \approx 0.8122 \times 10^2 = 81.2$ . The Earth is approximately $81$ times heavier.
$7 \times 10^{-5} \text{ m} = 70 \times 10^{-6} \text{ m} = 70\;\mu\text{m}$ .
$7.2 \times 4 = 28.8$ ; $10^{-3} \times 10^5 = 10^2$ . So $28.8 \times 10^2 = 2.88 \times 10^3$ .
Suppose $r$ is rational and $x$ is irrational and $r + x = q$ is rational. Then $x = q - r$ , a difference of two rationals, which is rational — contradicting $x$ being irrational.
$4.2^2 = 17.64$ and $4.3^2 = 18.49$ . Since $17.64 < 18 < 18.49$ , we have $4.2 < \sqrt{18} < 4.3$ .
$t = \dfrac{4.5 \times 10^{12}}{3 \times 10^8} = 1.5 \times 10^4 = 15\,000$ seconds (about $4.2$ hours).

Tier 3

Rational. $\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4$ .
$4.2 \times 10^{-7} \text{ m} = 420 \times 10^{-9} \text{ m} = 420$ nm.
$\sqrt{2} \times \sqrt{2} = 2$ , which is rational. This works because the definition says each individual number is irrational, not that products of irrationals must be irrational.
$\dfrac{9 \times 10^{11}}{2.6 \times 10^7} = \dfrac{9}{2.6} \times 10^4 \approx 3.46 \times 10^4 = 34\,615$ dollars per person.
Let $x = 0.\overline{9}$ . Then $10x = 9.\overline{9}$ , so $10x - x = 9$ , giving $9x = 9$ and $x = 1$ . This shows $0.\overline{9}$ and $1$ are the same number — every terminating decimal also has a repeating representation. It does not blur the rational/irrational boundary; both forms are rational.

Challenge

Assume $r + x = q$ where $q$ is rational. Then $x = q - r$ , a difference of two rationals, which is rational. This contradicts $x$ being irrational, so $r + x$ must be irrational.
$V = \dfrac{4}{3}\pi (4.4 \times 10^{26})^3 = \dfrac{4}{3}\pi \times 8.5184 \times 10^{79} \approx 4.19 \times 8.5184 \times 10^{79} \approx 3.6 \times 10^{80}$ m $^3$ .
Numerator: $(2.4)^2 \times (10^5)^2 = 5.76 \times 10^{10}$ . Division: $\dfrac{5.76 \times 10^{10}}{6 \times 10^3} = 0.96 \times 10^7 = 9.6 \times 10^6$ .
Operations per year: $3.6 \times 10^{12} \times 3.15 \times 10^7 = 11.34 \times 10^{19} = 1.134 \times 10^{20}$ . Audio processed: $1.134 \times 10^{20} \times 8 \times 10^{-9} = 9.072 \times 10^{11}$ seconds $= \dfrac{9.072 \times 10^{11}}{3600} \approx 2.52 \times 10^8$ hours.

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