Ordering Numbers and Using Inequality Symbols for the TMUA

Q: Is 11.0 greater than or equal to 11?

Yes. The symbol $\ge$ means 'greater than OR equal to'. Since $11.0$ is exactly equal to $11$, the condition is satisfied and $11.0 \ge 11$ is a true statement.

Q: How do I compare fractions with very large denominators?

If finding a common denominator is too time-consuming, convert each fraction to a decimal by dividing the numerator by the denominator. Usually, calculating to two or three decimal places is sufficient to determine the order.

Q: Why is -20,000 smaller than -235?

On a number line, $-20,000$ is much further to the left than $-235$. In the context of temperature or debt, $-20,000$ represents a 'lower' or 'more negative' state than $-235$.

Updated August 2025

Understanding how to order integers, decimals, and fractions is essential for success in university mathematics admissions tests. This guide covers the correct use of inequality symbols and provides systematic methods for comparing different numerical forms using place value analysis, common denominators, and conversion strategies.

Core concept

Numbers can be ordered by evaluating their placement on a number line or by comparing the place value of their digits. Relationships between these values are formally expressed using the symbols $=$ , $\neq$ , $<$ , $>$ , $\le$ , and $\ge$ .

Understanding Mathematical Symbols

In mathematics, specific symbols are used to describe the relationship between two quantities. You must be able to use and interpret the following:

$=$ is the symbol for 'is equal to'. For example, $6 + 4 = 10$ .
$\neq$ is the symbol for 'is not equal to'. For example, $6 + 4 \neq 11$ .
$<$ is the symbol for 'is less than'. For example, $6 + 4 < 11$ .
$>$ is the symbol for 'is greater than'. For example, $6 + 4 > 9$ .
$\le$ is the symbol for 'is less than or equal to'. This is true if the first number is either smaller than or exactly equal to the second. For example, $6 + 5 \le 11$ and $6 + 4 \le 11$ are both true.
$\ge$ is the symbol for 'is greater than or equal to'. This is true if the first number is either larger than or exactly equal to the second. For example, $6 + 5 \ge 11$ and $4 + 7 \ge 11$ are both true.

Worked Example: Using Symbols

If $x = 6$ and $y = -4$ , determine if the following statements are true or false:

$x + y > 4$ : Substituting the values gives $6 + (-4) = 2$ . Since $2$ is not greater than $4$ , this is false.
$x - y \neq 2$ : Substituting the values gives $6 - (-4) = 10$ . Since $10$ is not equal to $2$ , this is true.
$x + 4 \ge 9$ : Substituting the values gives $6 + 4 = 10$ . Since $10$ is greater than $9$ , this is true.
$3 - y = 5$ : Substituting the values gives $3 - (-4) = 7$ . Since $7$ is not equal to $5$ , this is false.

Ordering Integers and Decimals

Integers and decimals are ordered by comparing the place value of each of their digits. A common technique is to write the numbers in a column, ensuring the decimal points (and thus the place values) are aligned. You can also order numbers by their position on a number line. On a horizontal $x$ axis, larger numbers are found further to the right. On a vertical $y$ axis, larger numbers are higher up.

Worked Example: Ordering Integers

Write these integers in order of size, largest first: $89,340, 216,300, 789, -235, -1,356, -20,000, 99,567, -9,834$ .

Comparing the place values and the signs:

$216,300$ (largest)
$99,567$
$89,340$
$789$
$-235$ (smallest negative magnitude)
$-1,356$
$-9,834$
$-20,000$ (smallest)

Worked Example: Ordering Decimals

Write these decimals in order of size, largest first: $0.064, 0.00937, 0.1, 0.00876, 0.00098$ .

By comparing tenths, then hundredths, and so on:

$0.1$
$0.064$
$0.00937$
$0.00876$
$0.00098$

Ordering Fractions

Fractions can be compared by converting them to have a common denominator. Once the denominators are equal, the fraction with the larger numerator is the larger value. Alternatively, fractions can be converted into decimals or percentages for comparison.

Worked Example: Ordering Fractions

Write these fractions in order of size, largest first: $\frac{3}{5}, \frac{4}{10}, \frac{7}{15}, \frac{2}{3}, \frac{23}{30}$ .

The lowest common multiple of the denominators $(5, 10, 15, 3, 30)$ is $30$ . We convert each fraction:

$\frac{3}{5} = \frac{18}{30}$
$\frac{4}{10} = \frac{12}{30}$
$\frac{7}{15} = \frac{14}{30}$
$\frac{2}{3} = \frac{20}{30}$
$\frac{23}{30}$ remains $\frac{23}{30}$

Ordering the numerators from largest to smallest ( $23, 20, 18, 14, 12$ ), we get: $\frac{23}{30}, \frac{2}{3}, \frac{3}{5}, \frac{7}{15}, \frac{4}{10}$ .

Ordering a Mixture of Types

When a set contains a mix of integers, decimals, and fractions, it is usually easiest to convert all values into decimals or percentages. For fractions, divide the numerator by the denominator to the required number of decimal places.

Worked Example: Mixed Ordering

Write these numbers in order of size, smallest first: $\frac{3}{7}, 0.434, \frac{9}{20}, 0.0934, \frac{89}{1000}$ .

Convert all to decimals:

$\frac{3}{7} \approx 0.428...$
$0.434$
$\frac{9}{20} = 0.45$
$0.0934$
$\frac{89}{1000} = 0.089$

Ordering these decimals from smallest to largest gives $0.089, 0.0934, 0.428, 0.434, 0.45$ . Therefore, the final list is: $\frac{89}{1000}, 0.0934, \frac{3}{7}, 0.434, \frac{9}{20}$ .

Key takeaways

Inequality symbols always point towards the smaller number: $small < large$ or $large > small$ .
To order negative numbers, remember that the further from zero a negative number is, the smaller its value.
When comparing fractions, finding a common denominator allows for a direct comparison of the numerators.
In a mix of data types, converting everything to decimals is the most efficient strategy for comparison.

Tips

In the TMUA, you may face problems where you must compare expressions like $\sqrt{2}$ or $\pi$ . Always keep a few decimal approximations in mind: $\sqrt{2} \approx 1.41$ , $\sqrt{3} \approx 1.73$ , and $\pi \approx 3.14$ to help with quick ordering without a calculator.

Cautions

Be careful with negative signs when using inequality symbols. For example, while $10 > 5$ , when you multiply by $-1$ , the relationship reverses: $-10 < -5$ . Failing to reverse the sign when changing the polarity of an inequality is a common error.

Insight

The concept of 'greater than or equal to' forms the basis of optimization and boundary conditions in higher mathematics. It implies a 'closed' set in topology, where the boundary value itself is included in the set, unlike 'greater than' which implies an 'open' set.

Worked Examples

Example 1

a, b

and

c

are real numbers with

a < b < c < 0

Which of the following statements must be true?
I

ac < ab < a^2

b(c + a) > 0

III

\frac{c}{b} > \frac{a}{b}

A:none of them

B:I only

C:II only

D:III only

E:I and II only

F:I and III only

G:II and III only

H:I, II and III

Practice Questions

Practice Question 1

Which one of the following numbers is largest in value?

(All angles are given in radians.)

\tan(\frac{3\pi}{4})

\log_{10} 100

\sin^{10}(\frac{\pi}{2})

\log_2 10

(\sqrt{2}-1)^{10}

Frequently asked questions

Is 11.0 greater than or equal to 11?

Yes. The symbol $\ge$ means 'greater than OR equal to'. Since $11.0$ is exactly equal to $11$ , the condition is satisfied and $11.0 \ge 11$ is a true statement.

How do I compare fractions with very large denominators?

If finding a common denominator is too time-consuming, convert each fraction to a decimal by dividing the numerator by the denominator. Usually, calculating to two or three decimal places is sufficient to determine the order.

Why is -20,000 smaller than -235?

On a number line, $-20,000$ is much further to the left than $-235$ . In the context of temperature or debt, $-20,000$ represents a 'lower' or 'more negative' state than $-235$ .

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