0580 · 1.5Ordering

Ordering numbers and comparison symbols

0:00A temperature of negative two degrees is warmer than one of negative six. Yet six looks larger than two when we ignore their signs. Ordering numbers means comparing their actual values, not just the digits we see. By the end of this video, you will understand how to compare quantities and explain the symbols that describe their order. We need place value and the idea that fractions, decimals and percentages can represent the same amount.

0:24Ordering means arranging quantities according to their values. Ascending order runs from smallest to largest. Descending order is the reverse. Before comparing measurements, make sure they use the same units. A number line makes the central idea visible. Values increase as we move to the right, so any number is greater than those to its left. Negative two is therefore greater than negative six. Although negative six lies farther from zero, its value is smaller.

0:52The equals symbol states that two expressions have the same value. The not equal symbol states that their values differ, without saying which is larger. Greater than and less than compare two unequal values. Read every statement from left to right, keeping the order of the quantities unchanged. The pointed end of a comparison symbol faces the smaller value. Its wider opening faces the larger value because that side represents more.

1:16Eight is greater than three, and three is less than eight. These statements express the same comparison from opposite directions. Greater than or equal to allows two possibilities: a larger value or the same one. Less than or equal to permits a smaller value or the same one. For example, an age of at least sixteen means greater than or equal to sixteen. An age of at most sixteen means less than or equal to sixteen.

1:40The line underneath these symbols includes equality as an allowed case. Seven is greater than or equal to seven is true because the values are equal. Seven is greater than seven is false because neither value exceeds the other. Similarly, both five and seven are less than or equal to seven.

Ordering decimals and mixed forms

1:57For decimals, compare corresponding place values, starting on the left. The first different digit decides which positive decimal is larger. Trailing zeros after the decimal point do not change a value. Writing them can align the columns and make the comparison clearer. Our first example asks for these decimals in ascending order. Zero point seven becomes zero point seven zero zero. Zero point six eight becomes zero point six eight zero.

2:24Zero point seven zero five stays unchanged, while zero point zero seven becomes zero point zero seven zero. Compare the tenths digits first: seven, six, seven and zero. This puts zero point zero seven first and zero point six eight second. The remaining values have matching tenths and hundredths digits. Their thousandths digits are zero and five, so zero point seven comes before zero point seven zero five. The ascending order is zero point zero seven, zero point six eight, zero point seven, then zero point seven zero five.

2:58More decimal digits do not automatically mean a larger number. Each digit matters through its position. Descending order would reverse this completed list. Our second example mixes negative zero point four, three fifths, fifty-eight per cent and zero point six two. We compare them by expressing each in decimal form. Three fifths equals six tenths, which is zero point six zero. Fifty eight per cent means fifty-eight hundredths, giving zero point five eight.

3:27Negative zero point four comes first because it is less than every positive value in the list. Comparing hundredths gives zero point five eight, zero point six zero, then zero point six two. Using the original forms, the answer is negative zero point four, fifty-eight per cent, three fifths, then zero point six two. Conversion helps us compare, but we preserve the given quantities in our answer.

Comparing values in exam questions

3:51An exam question may ask you to insert equals, less than or greater than. To compare 3/4 with zero point seven five, express the fraction in hundredths. 3/4 = 75/100, which is zero point seven five. The correct symbol is equals. For negative five compared with negative two, use their number line positions. Negative five lies to the left of negative two. Therefore, negative five is less than negative two.

4:17Do not compare only five with two, because removing the signs changes the values. These methods must work without a calculator on Paper two. They remain useful on Paper four, where calculators are allowed. Avoid rounding before ordering, because rounding can make distinct values appear equal. Always check whether the question requests ascending or descending order. Finally, read your comparison aloud to check that the symbol points towards the smaller value.

4:43Ordering is about value, not appearance or distance from zero. Use a number line for signed numbers and matching place values for decimals. Convert mixed forms when that makes comparison easier. Equals means the same value, while the inclusive comparison symbols allow equality as well as an inequality.