Ordering Numbers and Mathematical Symbols for ESAT

Understanding Comparison Symbols

Mathematical symbols allow us to define the relationship between two values precisely. It is essential to distinguish between strict inequalities and those that include equality.

1. $=$ is the symbol for 'is equal to', for example: $6 + 4 = 10$.

2. $\neq$ is the symbol for 'is not equal to', for example: $6 + 4 \neq 11$.

3. $<$ is the symbol for 'is less than', for example: $6 + 4 < 11$.

4. $>$ is the symbol for 'is greater than', for example: $6 + 4 > 9$.

5. $\le$ is the symbol for 'is less than or equal to'. This statement is true if the first value is smaller than or equal to the second. For example: $6 + 5 \le 11$ is true, and $6 + 4 \le 11$ is also true.

6. $\ge$ is the symbol for 'is greater than or equal to'. For example: $6 + 5 \ge 11$ is true, and $6 + 7 \ge 11$ is also true.

Example: Testing Statements with Symbols

If we let $x = 6$ and $y = -4$, we can evaluate whether specific mathematical statements are true or false by substitution.

Is $x + y > 4$ true? Calculation: $x + y = 6 + (-4) = 2$. Since 2 is not greater than 4, the statement is false.

Is $x - y \neq 2$ true? Calculation: $x - y = 6 - (-4) = 6 + 4 = 10$. Since 10 is not equal to 2, the statement is true.

Is $x + 4 \ge 9$ true? Calculation: $x + 4 = 6 + 4 = 10$. Since 10 is greater than 9, the statement satisfies the 'greater than' part of the symbol and is therefore true.

Is $3 - y = 5$ true? Calculation: $3 - y = 3 - (-4) = 7$. Since 7 is not equal to 5, the statement is false.

Ordering Integers and Decimals

To order integers and decimals, we consider their sign (positive or negative) and compare the place value of their digits. A common technique is to align the numbers in a column so that the place values (units, tenths, hundredths, etc.) are maintained.

Alternatively, we can use a number line, often represented as an $x$-axis or $y$-axis. On a horizontal $x$-axis, larger numbers are positioned further to the right. On a vertical $y$-axis, larger numbers are positioned higher up.

Example: Ordering Large Integers

Consider the following list of integers to be ordered from largest to smallest: $89\ 340, 216\ 300, 789, -235, -1356, -20\ 000, 99\ 567, -9834$.

By comparing the place values of the positive numbers and the magnitude of the negative numbers, we obtain the order: $216\ 300, 99\ 567, 89\ 340, 789, -235, -1356, -9834, -20\ 000$.

Example: Ordering Decimal Values

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

By comparing the tenths column, we see $0.1$ is the largest. Comparing the hundredths column, $0.064$ follows. Continuing this process, we get: $0.1, 0.064, 0.00937, 0.00876, 0.00098$.

Ordering Fractions

Fractions can be ordered by two primary methods:

1. Converting all fractions to a common denominator and then comparing their numerators.
2. Converting all fractions into decimals or percentages to compare their magnitudes directly.

Example: Using a Common Denominator

Order these fractions from largest to smallest: $\frac{3}{5}, \frac{4}{10}, \frac{7}{15}, \frac{2}{3}, \frac{23}{30}$.

All of the denominators are factors of 30. We can write each as an equivalent fraction with a denominator of 30: $\frac{3}{5} = \frac{18}{30}$ $\frac{4}{10} = \frac{12}{30}$ $\frac{7}{15} = \frac{14}{30}$ $\frac{2}{3} = \frac{20}{30}$ $\frac{23}{30}$ remains the same.

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

Ordering a Mixture of Formats

When presented with a mix of integers, decimals, and fractions, it is usually easiest to convert every value into a decimal or a percentage for a direct comparison.

Example: Mixed Numbers

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

Convert the fractions to decimals by dividing the numerator by the denominator: $\frac{3}{7} = 0.428...$ $0.434$ $\frac{9}{20} = 0.45$ $0.0934$ $\frac{89}{1000} = 0.089$

Comparing these decimals from smallest to largest, we get $0.089, 0.0934, 0.428, 0.434, 0.45$. Converting back to the original forms, the list is: $\frac{89}{1000}, 0.0934, \frac{3}{7}, 0.434, \frac{9}{20}$.