Standard form, surds and number systems

Q: What is the difference between rational and irrational numbers?

A rational number can be written as a fraction \(\frac{a}{b}\) of two integers (terminating or recurring as a decimal), whereas an irrational number cannot and its decimal goes on forever without repeating.

Q: How do you convert a recurring decimal to an exact fraction?

Set \(x\) equal to the decimal, multiply by powers of \(10\) to align the repeating parts, subtract the equations to eliminate the repeating digits, and solve for \(x\).

Q: What is the correct structure of a number written in standard form?

A number in standard form is written as \(A \times 10^n\), where \(1 \le A

Q: How do you convert ordinary numbers into standard form?

Place the decimal point after the first non-zero digit to find \(A\), then count the number of places the decimal moved to find the power \(n\) (positive for large numbers, negative for small decimals).

Q: How do you multiply and divide numbers written in standard form?

Multiply or divide the front numbers (\(A\)), then add (for multiplication) or subtract (for division) the powers of \(10\), adjusting the answer so \(1 \le A

Q: What is a surd?

A surd is an irrational root (such as a square root) that cannot be written as a whole number or simple fraction.

Q: How do you simplify a square root surd?

Find the largest square factor of the number under the root, split the root using \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), and evaluate the square root of the square factor.

Q: When can surds be added or subtracted together?

Surds can only be added or subtracted if they have the exact same number under the root (like surds), often after simplifying each term first.

Q: How do you expand brackets containing surds?

Use standard algebraic expansion rules (like FOIL), multiplying terms together and using \(\sqrt{a} \times \sqrt{a} = a\) and \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\).

Q: How do you rationalise the denominator of a fraction with a single surd \(\frac{a}{\sqrt{b}}\)?

Multiply both the numerator and the denominator by \(\sqrt{b}\) to eliminate the root in the denominator.

Mathematics 2210 · CCEA GCSE · 8 min read · Updated: Aug 31, 2026

Welcome to Number Systems, Standard Form, and Surds!

Numbers are the foundation of all mathematics. In this chapter, we explore how different types of numbers are grouped, how to write huge numbers (like the distance to stars) or tiny numbers (like the size of a cell) using Standard Form, and how to work with Surds so our answers stay 100% exact without messy rounding. Don't worry if some of this seems new or challenging at first — we will break down each topic step by step!

Part 1: Number Systems (Classifying Numbers)

Just like animals can be sorted into mammals, birds, and reptiles, numbers belong to different families depending on their properties.

1. The Main Sets of Numbers

Natural Numbers (\(\mathbb{N}\)): These are the positive counting numbers starting from \(1\).
Examples: \(1, 2, 3, 4, 5, \dots\)

Integers (\(\mathbb{Z}\)): All positive and negative whole numbers, including zero. They have no fractional or decimal parts.
Examples: \(-4, -1, 0, 3, 12\)

Rational Numbers (\(\mathbb{Q}\)): Any number that can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\). When written as decimals, rational numbers either terminate (end) or recur (repeat a pattern).
Examples: \(\frac{3}{4} = 0.75\) (terminating), \(\frac{1}{3} = 0.333\dots = 0.\dot{3}\) (recurring), \(-5 = \frac{-5}{1}\), \(7 = \frac{7}{1}\)

Irrational Numbers: Numbers that cannot be written as a simple fraction of two integers. As decimals, they go on forever without ever repeating in a regular pattern.
Examples: \(\pi = 3.14159\dots\), \(\sqrt{2} = 1.41421\dots\), \(\sqrt{7} = 2.64575\dots\)

Real Numbers (\(\mathbb{R}\)): The entire collection of all rational and irrational numbers put together. Any number you can find on a standard number line is a real number.

2. Converting Recurring Decimals to Fractions

Because recurring decimals are rational numbers, we can convert them into exact fractions using algebra.

Step-by-Step Method:
1. Set \(x\) equal to the recurring decimal.
2. Multiply \(x\) by a power of \(10\) (\(10, 100, 1000\), etc.) to shift one full repeating cycle in front of the decimal point.
3. Subtract the original equation from the new equation to cancel out the repeating tail.
4. Solve for \(x\) and simplify your fraction.

Worked Example 1: Convert \(0.\dot{7}\) to a fraction.
Let \(x = 0.7777\dots\)
Multiply by \(10\):
\(10x = 7.7777\dots\)
Subtract the equations: \((10x - x = 7.7777\dots - 0.7777\dots)\)
\(9x = 7\)
Divide by \(9\):
\(x = \frac{7}{9}\)

Worked Example 2: Convert \(0.2\dot{3}\dot{5}\) to a fraction.
Let \(x = 0.2353535\dots\)
Multiply by \(10\) to get non-repeating digits out of the way: \(10x = 2.353535\dots\)
Multiply by \(1000\) to move one repeating cycle: \(1000x = 235.353535\dots\)
Subtract: \(1000x - 10x = 235.353535\dots - 2.353535\dots\)
\(990x = 233\)
\(x = \frac{233}{990}\)

Common Mistake to Avoid:
Do not assume \(\sqrt{9}\) is irrational just because it has a square root symbol! Since \(\sqrt{9} = 3\), it is an integer and a rational number.

Key Takeaway for Number Systems:
Every number on a number line is a Real Number. If it can be written as a fraction, it is Rational; if its decimal goes on forever without repeating, it is Irrational.

Part 2: Standard Form (Scientific Notation)

Standard form is a quick, neat way to write very large or very small numbers using powers of \(10\).

1. The Rule of Standard Form

A number written in standard form always looks like this:

\(A \times 10^n\)

Where:
• \(A\) must be between \(1\) and \(10\) (meaning \(1 \le A < 10\)). It can be \(1\), but it cannot be \(10\).
• \(n\) must be an integer (a positive or negative whole number).

Memory Aid:
• Positive power of 10 (\(10^n\)): The number is large (\(\ge 10\)).
• Negative power of 10 (\(10^{-n}\)): The number is tiny (between \(0\) and \(1\)).

2. Converting Between Ordinary Numbers and Standard Form

Converting Large Numbers:
Convert \(450,000\) into standard form:
1. Place the decimal point after the first non-zero digit to get \(A\): \(4.5\)
2. Count how many places the decimal moved from the end to where it is now: \(5\) places to the left.
3. Answer: \(4.5 \times 10^5\)

Converting Small Numbers:
Convert \(0.00038\) into standard form:
1. Place the decimal point after the first non-zero digit: \(3.8\)
2. Count how many places the decimal moved to the right: \(4\) places.
3. Since it is a tiny decimal, the power is negative: \(3.8 \times 10^{-4}\)

3. Calculations in Standard Form

Multiplication: Multiply the front numbers and add the powers of \(10\).
Example: Calculate \((3 \times 10^4) \times (2 \times 10^5)\)
\((3 \times 2) \times (10^4 \times 10^5) = 6 \times 10^{4+5} = 6 \times 10^9\)

Adjusting if \(A \ge 10\):
\((4 \times 10^5) \times (5 \times 10^3) = 20 \times 10^8\)
Since \(20\) is not between \(1\) and \(10\), rewrite \(20\) as \(2.0 \times 10^1\):
\(2.0 \times 10^1 \times 10^8 = 2.0 \times 10^9\)

Division: Divide the front numbers and subtract the powers of \(10\).
Example: Calculate \((8 \times 10^7) \div (2 \times 10^3)\)
\((8 \div 2) \times (10^7 \div 10^3) = 4 \times 10^{7-3} = 4 \times 10^4\)

Adjusting if \(A < 1\):
\((2 \times 10^4) \div (8 \times 10^6) = 0.25 \times 10^{-2}\)
Rewrite \(0.25\) as \(2.5 \times 10^{-1}\):
\(2.5 \times 10^{-1} \times 10^{-2} = 2.5 \times 10^{-3}\)

Addition and Subtraction:
Make sure the powers of \(10\) match before adding or subtracting, or convert to ordinary numbers first.
Example: Calculate \((4.3 \times 10^4) + (2.1 \times 10^3)\)
Convert to ordinary numbers: \(43,000 + 2,100 = 45,100\)
Convert back to standard form: \(4.51 \times 10^4\)

Common Mistake to Avoid:
Writing \(45 \times 10^3\) or \(0.7 \times 10^5\). These are not in standard form because the front number is not between \(1\) and \(10\)!

Key Takeaway for Standard Form:
Always check that your final answer has a front value \(1 \le A < 10\). If it doesn't, adjust the power of \(10\) accordingly.

Part 3: Surds (Working with Exact Roots)

1. What is a Surd?

A surd is a root (like a square root) that cannot be simplified into a whole number or a simple fraction. For example, \(\sqrt{4} = 2\) is not a surd, but \(\sqrt{2} = 1.41421356\dots\) is a surd.
We keep numbers written as surds to give an exact answer rather than rounding off to decimal places.

2. Key Rules of Surds

• Multiplication Rule: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\)
• Division Rule: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)
• Squaring Rule: \(\sqrt{a} \times \sqrt{a} = (\sqrt{a})^2 = a\)

3. Simplifying Surds

To simplify a surd, look for the largest square number (\(4, 9, 16, 25, 36, 49, 64, 81, 100, \dots\)) that divides into the number under the square root.

Worked Example 1: Simplify \(\sqrt{50}\)
1. Find factors where one is a square number: \(50 = 25 \times 2\)
2. Split the root: \(\sqrt{50} = \sqrt{25} \times \sqrt{2}\)
3. Evaluate the square root: \(\sqrt{25} = 5\)
4. Combine: \(5\sqrt{2}\)

Worked Example 2: Simplify \(\sqrt{72}\)
• The largest square factor of \(72\) is \(36\).
\(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\)

4. Adding and Subtracting Surds

Surds can only be added or subtracted if they have the same root (just like collecting like terms in algebra, e.g., \(3x + 2x = 5x\)).

Example: Simplify \(\sqrt{12} + \sqrt{27}\)
Step 1: Simplify each surd first.
\(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\)
\(\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}\)
Step 2: Collect the like surds.
\(2\sqrt{3} + 3\sqrt{3} = (2 + 3)\sqrt{3} = 5\sqrt{3}\)

5. Expanding Brackets with Surds

We expand brackets with surds just like standard algebraic brackets (using FOIL or the grid method).

Single Brackets:
\(\sqrt{3}(4 + \sqrt{3}) = (\sqrt{3} \times 4) + (\sqrt{3} \times \sqrt{3}) = 4\sqrt{3} + 3\)

Double Brackets:
Expand and simplify \((3 + \sqrt{2})(5 - \sqrt{2})\):
• First: \(3 \times 5 = 15\)
• Outside: \(3 \times (-\sqrt{2}) = -3\sqrt{2}\)
• Inside: \(\sqrt{2} \times 5 = 5\sqrt{2}\)
• Last: \(\sqrt{2} \times (-\sqrt{2}) = -2\)
Combine terms: \((15 - 2) + (-3\sqrt{2} + 5\sqrt{2}) = 13 + 2\sqrt{2}\)

6. Rationalising the Denominator

In mathematics, it is standard practice not to leave a surd in the denominator (bottom) of a fraction. Rationalising means making the denominator a rational number (a whole number).

Type 1: Single Term Denominators (\(\frac{a}{\sqrt{b}}\))
Multiply the numerator and denominator by \(\sqrt{b}\):
Example: Rationalise \(\frac{6}{\sqrt{3}}\)
\(\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3}\)
Simplify the whole numbers: \(\frac{6}{3}\sqrt{3} = 2\sqrt{3}\)

Type 2: Binomial Denominators (\(\frac{a}{b + \sqrt{c}}\) or \(\frac{a}{b - \sqrt{c}}\))
Multiply top and bottom by the conjugate (change the middle sign). This uses the difference of two squares: \((a + b)(a - b) = a^2 - b^2\).

Example: Rationalise \(\frac{4}{3 - \sqrt{5}}\)
1. Multiply top and bottom by \((3 + \sqrt{5})\):
\(\frac{4(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})}\)
2. Expand the top: \(4 \times 3 + 4 \times \sqrt{5} = 12 + 4\sqrt{5}\)
3. Expand the bottom: \(3^2 - (\sqrt{5})^2 = 9 - 5 = 4\)
4. Simplify: \(\frac{12 + 4\sqrt{5}}{4} = \frac{12}{4} + \frac{4\sqrt{5}}{4} = 3 + \sqrt{5}\)

Common Mistake to Avoid:
Thinking that \(\sqrt{a + b} = \sqrt{a} + \sqrt{b}\). This is not true! For example, \(\sqrt{9 + 16} = \sqrt{25} = 5\), but \(\sqrt{9} + \sqrt{16} = 3 + 4 = 7\).

Key Takeaway for Surds:
Always look for square factors to simplify surds, treat identical surds like like terms in algebra, and eliminate square roots from denominators by multiplying by an appropriate form of \(1\).

Quick Summary Checklist

Before finishing this chapter, make sure you can:
• Identify whether a number is natural, integer, rational, or irrational.
• Convert recurring decimals into fractions using algebra.
• Write large and small numbers in standard form \(A \times 10^n\).
• Add, subtract, multiply, and divide numbers in standard form.
• Simplify surds by finding square factors.
• Expand brackets and collect like surds.
• Rationalise single-term and two-term denominators.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the difference between rational and irrational numbers?

A rational number can be written as a fraction \(\frac{a}{b}\) of two integers (terminating or recurring as a decimal), whereas an irrational number cannot and its decimal goes on forever without repeating.

How do you convert a recurring decimal to an exact fraction?

Set \(x\) equal to the decimal, multiply by powers of \(10\) to align the repeating parts, subtract the equations to eliminate the repeating digits, and solve for \(x\).

What is the correct structure of a number written in standard form?

A number in standard form is written as \(A \times 10^n\), where \(1 \le A < 10\) and \(n\) is an integer.

How do you convert ordinary numbers into standard form?

Place the decimal point after the first non-zero digit to find \(A\), then count the number of places the decimal moved to find the power \(n\) (positive for large numbers, negative for small decimals).

How do you multiply and divide numbers written in standard form?

Multiply or divide the front numbers (\(A\)), then add (for multiplication) or subtract (for division) the powers of \(10\), adjusting the answer so \(1 \le A < 10\).

What is a surd?

A surd is an irrational root (such as a square root) that cannot be written as a whole number or simple fraction.

How do you simplify a square root surd?

Find the largest square factor of the number under the root, split the root using \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), and evaluate the square root of the square factor.

When can surds be added or subtracted together?

Surds can only be added or subtracted if they have the exact same number under the root (like surds), often after simplifying each term first.

How do you expand brackets containing surds?

Use standard algebraic expansion rules (like FOIL), multiplying terms together and using \(\sqrt{a} \times \sqrt{a} = a\) and \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\).

How do you rationalise the denominator of a fraction with a single surd \(\frac{a}{\sqrt{b}}\)?

Multiply both the numerator and the denominator by \(\sqrt{b}\) to eliminate the root in the denominator.

How do you rationalise a binomial denominator of the form \(\frac{a}{b - \sqrt{c}}\) or \(\frac{a}{b + \sqrt{c}}\)?

Multiply numerator and denominator by the conjugate (changing the sign between terms) using the difference of two squares: \((b - \sqrt{c})(b + \sqrt{c}) = b^2 - c\).

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