Ordering

The order symbols

• $=$ means is equal to, and $\ne$ means is not equal to.
• $<$ means is less than, and $>$ means is greater than.
• $⩽$ means is less than or equal to, and $⩾$ means is greater than or equal to.

The wide, open end of $<$ or $>$ always faces the larger number. $3<8$ and $8>3$ say the same thing, read from opposite ends. In the same way $-2<x$ means exactly the same as $x>-2$.

The strict signs $<$ and $>$ leave the boundary value out; $⩽$ and $⩾$ include it. If you must be at least 18 to vote, the ages allowed are $a⩾18$, because 18 itself counts.

Turning words into symbols:

• at least, no less than, a minimum of: $⩾$
• at most, no more than, a maximum of: $⩽$
• more than, over, above: $>$
• less than, fewer than, under, below: $<$

Ordering integers and negative numbers

On a number line, numbers increase from left to right. Every number is greater than every number to its left.

For negative numbers this means the bigger the number after the minus sign, the smaller the number. $-5<-2$, even though $5>2$, because $-5$ is further below zero. A temperature of $-5^{°}$C is colder than $-2^{°}$C.

In order from smallest: $-7,-4,-1,0,2$.

Ordering decimals

Line the numbers up by their decimal points and compare digit by digit from the left: units first, then tenths, then hundredths, and so on. The first place where the digits differ decides the order.

It helps to write in zeros so that every number has the same number of decimal places:

• $0.42\to 0.420$
• $0.402$
• $0.4\to 0.400$
• $0.425$

Now $0.400<0.402<0.420<0.425$. A decimal with more digits is not always larger: $0.402<0.42$, because in the hundredths place 0 is less than 2.

Ordering fractions

With the same denominator, compare the numerators: $\frac{5}{9}>\frac{4}{9}$.

With the same numerator, the larger denominator gives the smaller fraction, because the whole is cut into more pieces: $\frac{1}{9}<\frac{1}{8}$.

Otherwise, rewrite every fraction with a common denominator, usually the LCM of the denominators, and compare the numerators. Or change each fraction to a decimal. Both work; the common denominator method keeps the numbers exact, which is safer without a calculator.

Numbers in different forms

When a list mixes fractions, decimals and percentages, change them all into the same form, usually decimals. Then order the decimals, and write the answer using the original numbers, as the question gave them.

Work to enough decimal places to separate the numbers: three or four is usually plenty. Keep a note of which decimal came from which number, so the final list is easy to write.

Quantities with units

Quantities can only be compared in the same unit. Change them all to one unit first, then order them. For example $1.25$ km, $1300$ m and $124000$ cm become $1250$ m, $1300$ m and $1240$ m, so the shortest is $124000$ cm even though it has the biggest number.

The conversions you need most: $1$ km $=1000$ m, $1$ m $=100$ cm, $1$ cm $=10$ mm, $1$ kg $=1000$ g.

Using a calculator to order

Roots, $\pi$, powers and fractions with awkward denominators are easiest to compare with a calculator. Write each one as a decimal with enough places to tell them apart:

• $\sqrt{10}=3.1623\dots$
• $\pi =3.1416\dots$
• $\frac{22}{7}=3.1429\dots$

So $\pi <\frac{22}{7}<\sqrt{10}$. Two decimal places would not have been enough: $\pi$ and $\frac{22}{7}$ both round to 3.14.

Powers with negative indices are small positive numbers: $2^{-3}=\frac{1}{8}=0.125$. Numbers in standard form can be compared by their powers of 10 first: $3\times {10}^{-2}>9\times {10}^{-3}$, because ${10}^{-2}$ is ten times ${10}^{-3}$.