**Ordering Numbers**

When ordering numbers, they can either go in **ascending order** (smallest to largest) or **descending order** (largest to smallest). There are **5** main ways that you will be asked to order numbers.

Make sure you are happy with the following topics before continuing.

**Type 1: Ordering decimals**

You may be asked to order decimals, to do this we need to look at each number in turn and compare starting with the first digit.

**Example:** Put the following numbers in **ascending order**.

1.101, \,\,\, 0.11, \,\,\, 1.10, \,\,\, 0.01

**Step 1: ** look at the first digit for each number and compare.

\xcancel{\textcolor{blue}{1}.101}, \,\,\, \overset{\downarrow}{\textcolor{blue}{0}}.11, \,\,\, \xcancel{\textcolor{blue}{1}.10}, \,\,\, \overset{\downarrow}{\textcolor{blue}{0}}.01

We can see 2 numbers start with 1 and 2 numbers start with 0, as we are looking for the smallest first, we can eliminate the two largest.

**Step 2:** look at the second digits for the remaining numbers.

\xcancel{0.\textcolor{blue}{1}1}, \,\,\, 0.\overset{\downarrow}{\textcolor{blue}{0}}1

We can see that the second digit for the 1st number is a 1 and for the 2nd number it is a 0, so as we are looking for the smallest first, we can eliminate the 1st number.

So we have 0.01 in position 1 and 0.11 in position 2.

**Step 3:** look at the other numbers that we previously crossed out, to see which of them is the smallest.

\xcancel{1.{\textcolor{blue}{1}} {\textcolor{red}{0}}{\textcolor{limegreen}{1}}}, \,\,\, 1.{\textcolor{blue}{1}}{\textcolor{red}{0}}

We look at the second digit for each number and compare, but these are the same. Comparing the 3rd digit, but again these are the same. We can see that the 1st number has a fourth digit, but the 2nd number doesn’t, so we can eliminate the 1st number as it is larger.

So we have 1.10 in position 3 and 1.101 in position 4.

**Step 4:** write down the correct order, which is

0.01, \,\,\, 0.11, \,\,\, 1.10, \,\,\, 1.101

**Type 2: Ordering Large Numbers**

You may be asked to order large numbers, to do this we need to look at each number in turn and compare starting with the first digit.

**Example:** Put the following numbers in **descending order**.

3456, \,\,\, 5364, \,\,\, 3465, \,\,\ 5634

**Step 1:** we look at the first digit for each number and compare.

\xcancel{\textcolor{blue}{3}456}, \,\,\, \overset{\downarrow}{\textcolor{blue}{5}}364, \,\,\, \xcancel{\textcolor{blue}{3}465}, \,\,\, \overset{\downarrow}{\textcolor{blue}{5}}634

2 of the number start with 3. As we are looking for the largest first, we can eliminate the 2 smallest.

**Step 2:** look at the second digits.

5 \xcancel{\textcolor{blue}{3}64}, \,\,\, 5\overset{\downarrow}{\textcolor{blue}{6}}34

We know 6>3 so we can eliminate the first number.

So we have 5634 in position 1 and 5364 in position 2.

**Step 3:** look at the other numbers.

3 \xcancel{\textcolor{blue}{4}\textcolor{red}{5}6}, \,\,\, 3 \textcolor{blue}{4}\overset{\downarrow}{\textcolor{red}{6}} 5

The second digit for each number is the same, so we compare the third digit. We know 6>5 so we can eliminate the first number.

So we have 3465 in position 3 and 3456 in position 4.

**Step 4:** write down the correct order, which is

5634, \,\,\, 5364, \,\,\, 3465, \,\,\, 3456

**Type 3: Ordering Fractions, Decimals and Percentages**

We may be asked to order fractions, decimals and percentages, which can be tricky. Revise fractions, decimals and percentages here.

**Example:** Put the following numbers in **ascending order**.

\dfrac{1}{5}, \,\,\, 21 \%, \,\,\, 0.205, \,\,\, \dfrac{1}{4}

**Step 1: **First, we need to convert all of these numbers into the same format. For this example we will convert them all to decimals.

0.205 is already in decimal form.

Fractions to decimals:

\dfrac{1}{5} = 0.2 \,\,\, and \,\,\, \dfrac{1}{4} = 0.25

Percentage to a decimal:

21 \% \div 100 = 0.21

**Step 2: **We now rewrite the list using our conversions to decimals:

0.2, \,\,\, 0.21, \,\,\, 0.205, \,\,\, 0.25

**Step 3: **We now use what we have already learnt from Type 1 to order our list in ascending order.

0.2, \,\,\, 0.205, \,\,\, 0.21, \,\,\, 0.25

**Step 4: **Finally, write down the correct order, **converting back to the original values**, which is

\dfrac{1}{5}, \,\,\, 0.205, \,\,\, 0.21\%, \,\,\, \dfrac{1}{4}

**Type 4: Ordering Negative Numbers**

Remember that for negative numbers, the **bigger** the number after the minus sign, the **smaller** the value of the negative number.

**Example: **Put the following numbers in **descending order**.

2.3, \,\,\, -2.3, \,\,\, 3.2, \,\,\, -3.2

**Step 1:** We first look at the positive numbers and compare their first digits.

\xcancel{\textcolor{blue}{2}.3}, \,\,\, \overset{\downarrow}{\textcolor{blue}{3}}.2

3>2, this gives 3.2 in position 1 and 2.3 in position 2.

**Step 2:** We now look at the negative numbers and compare their first digits.

- \overset{\downarrow}{\textcolor{blue}{2}}.3, \,\,\, - \xcancel{\textcolor{blue}{3}.2}

-3 is **more negative** than - 2

So we have -2.3 in position 3 and -3.2 in position 4.

**Step 3:** Finally, we write down the correct order, which is

3.2, \,\,\, 2.3, \,\,\, -2.3, \,\,\, -3.2

**Type 5: Ordering Algebraic Terms**

We may also be asked to order numbers that are algebraic, in terms of x for example, with a given condition on its value.

**Example: **Put the following numbers in **ascending order**, given that x is a positive integer greater than 1.

x^2, \,\,\, \dfrac{1}{x}, \,\,\, x, \,\,\, x+1

**Step 1: **Substitute a value in for x that is greater than 1. We will substitute in x=2 as this will be easiest.

x^2 = 2^2 = 4 \,\,\,, \,\,\, \dfrac{1}{x} = \dfrac{1}{2}\,\,\,, \,\,\, x = 2\,\,\,, \,\,\, x+1 = 2+1 = 3

**Step 2:** We now rewrite the list using the substituted values.

4, \,\,\, \dfrac{1}{2}, \,\,\, 2, \,\,\, 3

**Step 3: **We now use what we have already learnt to order our list in ascending order.

\dfrac{1}{2}, \,\,\, 2, \,\,\, 3, \,\,\, 4

**Step 4: **Finally, write down the correct order, **writing in terms of** x, which is

\dfrac{1}{x}, \,\,\, x, \,\,\, x+1, \,\,\, x^2

**Example 1: Ascending Order**

Write these numbers in **ascending order** of size.

460,\,\,\, 346,\,\,\,64,\,\,\,46,\,\,\,476

**[1 mark]**

We need to start with the smallest and increase in size.

Now, the biggest numbers here are hundreds.

460,\,\,\, 346\,\,\,476

Then, numbers in the 400‘s

460<476

This gives the final answer to be,

46,\,\,\,64,\,\,\,346,\,\,\,460,\,\,\,476

**Example 2: Descending Order**

Write these numbers in **descending order** of size.

-3,\,\,\,12,\,\,\,18,\,\,\,-21,\,\,\,4

**[1 mark]**

Firstly, deal with the positives. Going in **descending order** – from largest to smallest – we get

18,\,\,\,12,\,\,\,4

Now, recall that negative numbers get smaller when the number after the minus sign gets bigger.

The completed list is

18,\,\,\,12,\,\,\,4,\,\,\,-3,\,\,\,-21

**Example 3: Ascending Order**

Write the following decimals in order from** smallest to largest**.

0.6,\,\,\,0.31,\,\,\,0.07,\,\,\,1.04,\,\,\,0.998

**[1 mark]**

Firstly, 1.04 is the only value bigger than 1 here so it must be the largest.

Secondly, 0.07 is the only value that’s smaller than 0.1, so it must be the smallest.

Continuing this we get,

0.07,\,\,\,0.31,\,\,\,0.6,\,\,\,0.998,\,\,\,1.04

**Example 4: Ordering Numbers of Different Types**

Put the numbers listed below in order from **smallest to largest**.

\dfrac{2}{5},\,\,\,0.45,\,\,\,44.5\%,\,\,\,\dfrac{7}{20}

**[2 marks]**

In this example, we will convert all numbers to decimals.

0.45 is already in decimal form.

44.5\% =44.5\div 100 = 0.445

\dfrac{2}{5} = 0.4

\dfrac{7}{20} = \dfrac{7\times 5}{20\times 5}=\dfrac{35}{100}=35\div 100 = 0.35

Now we have our 4 decimals, we can put them in order from smallest to largest:

0.35,\,\,\,0.4,\,\,\,0.445,\,\,\,0.45

Finally, we must write the numbers in order in their original forms. This looks like

\dfrac{7}{20},\,\,\,\dfrac{2}{5},\,\,\,44.5\%,\,\,\,0.45

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### Example Questions

**Question 1:** Write the following numbers in descending order:

-42,\,\,\,23,\,\,\,-23.5,\,\,\,1,\,\,\,4,\,\,\,

**[1 mark]**

**Descending** order means from **largest to smallest**, so let’s first place the positive numbers in order,

23,\,\,\,4,\,\,\,1

Recall that for negative numbers, the bigger the number after the minus sign, the smaller the value of the negative number.

42 is bigger than 23.5, which means -42 is smaller than -23.5, so the order should be,

23,\,\,\,4,\,\,\,1,\,\,\,-23.5,\,\,\,-42

**Question 2:** Write the following numbers in ascending order:

2.5,\,\,\,2.04,\,\,\,2.58,\,\,\,3.5,\,\,\,2.8

**[1 mark]**

**Ascending** order means from **smallest to largest**, hence the correct order is,

2.04,\,\,\,2.5,\,\,\,2.58,\,\,\,2.8,\,\,\,3.5

**Question 3:** Order the following numbers from smallest to largest:

5.23,\,\,\,\,\,5.12,\,\,\,\,\,\,4.092,\,\,\,\,\,5.01,\,\,\,\,\,4.87

**[1 mark]**

Ordering decimals requires comparing digits in the same columns, starting with the digits in the **place value column **that is furthest to the left, hence the correct order is,

4.092,\,\,\,\,\,4.87,\,\,\,\,\,5.01,\,\,\,\,\,5.12,\,\,\,\,\, 5.23

**Question 4:** Write the following in order from smallest to largest:

64\%,\,\,\,\,\,0.633,\,\,\,\,\,\dfrac{5}{8},\,\,\,\,\,64.4\%

**[2 marks]**

To compare the sizes of these numbers, we need to have them all in the same form.

In order to turn a percentage into a decimal, we divide by 100, hence,

64\%=64\div 100=0.64

and,

64.4\%=64.4\div 100=0.644

We can also convert a fraction to a decimal,

\dfrac{5}{8} = 0.625

Placing the values in order then,

0.625,\,\,\,0.633,\,\,\,0.64,\,\,\,0.644

Finally, putting them in order in the original forms, we get

\dfrac{5}{8},\,\,\,0.633,\,\,\,64\%,\,\,\,64.4\%

**Question 5:** Write the following numbers in ascending order, given that x is a positive integer greater than 2.

\dfrac{1}{x} ,\,\,\,\,\,x^2,\,\,\,\,\,\,x,\,\,\,\,\,(x+1),\,\,\,\,\,2x

**[2 marks]**

A simple way to compare the sizes of these numbers is to substitute a value in for x that is greater than 2.

We will substitute in x=3 for each of the terms,

\dfrac{1}{x}=\dfrac{1}{3}

x^2=3^2=9

x=3

(x+1)=4

2x=6

Now it is straightforward to place them in order,

\dfrac{1}{3} ,\,\,\,\,\,3,\,\,\,\,\,\,4,\,\,\,\,\,6,\,\,\,\,\,9

Hence we can place the original terms in the same corresponding order,

\dfrac{1}{x} ,\,\,\,\,\,x,\,\,\,\,\,\,(x+1),\,\,\,\,\,2x,\,\,\,\,\,x^2

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