Decimals (I): Introduction

1Summary 重點摘要

0.5 is half. First digit after dot = tenths.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: What decimals areDecimals show numbers less than 1. Split a 1-metre ruler into 10 equal parts, and each part is 0.1 m, which is one tenth and is written with a decimal point. Parts that are less than 1, written with the point, are decimals; 0.1 m is 10 centimetres, and 1 centimetre is 0.01 m. A common mistake is to think of a decimal as 'a smaller whole number', or to think 0.1 is smaller than 0.05. Remember that 0.1 is one tenth and 0.05 is five hundredths, and 0.1 = 0.10, which is larger than 0.05; comparing decimals is not a matter of counting digits.
2. Part 2: What the point doesThe decimal point separates the whole part from the decimal part. In 1.2, the 1 sits to the left of the point and means 1 whole; the 2 sits in the first place to the right and means 2 tenths. Left of the point are wholes; right of it are parts of one. For example, 3.7 means 3 wholes plus 7 tenths. A common mistake is to misplace the point, writing 1.2 as 12, which is ten times too big. The point is the border between wholes and parts, and its position may not shift by even one place. When reading a decimal, read the whole part first, then the word 'point', then the decimal digits one by one.
3. Part 3: Tenths and hundredthsThe first place right of the point is the tenths place, and the second is the hundredths place. 0.4 is 4 tenths; 0.45 is 4 tenths plus 5 hundredths, that is 0.45 = 0.4 + 0.05. The further right a place sits, the smaller it is: one tenth equals ten hundredths, so 0.5 = 0.50. A common mistake is to think 0.45 is bigger than 0.5 because it has one more digit; in fact the tenths digit 5 beats the tenths digit 4, so 0.5 > 0.45. Compare the tenths place first, and only if they are equal move to the hundredths, going from left to right one place at a time.
4. Part 4: How to read themDecimals are read in a fixed way: 0.1 is read as zero point one, 0.5 as zero point five, 1.8 as one point eight, and 12.6 as twelve point six. The dot is read as point, and the digits after it are read one by one, not as tens; remember this rule. A common mistake is to read 0.45 as 'zero point forty-five'; the correct reading is 'zero point four five', digit by digit. Another is to skip the 'zero' when the whole part is 0. The rule for reading decimals: read the whole part as usual, then read the digits after the point one at a time, one digit one sound, no more and no less.
5. Part 5: Decimals and fractions are cousinsDecimals and fractions are close cousins: 0.3 = 310, 0.5 = 510, which is exactly one half, and 0.7 = 710. Any number of tenths can be written as a decimal with one place after the point, so the two forms swap easily: 810 becomes 0.8, and 0.9 becomes 910. A common mistake is to confuse the denominator in the swap, writing 0.3 as 3100; the tenths place matches a denominator of 10, and only the hundredths place matches 100, so 0.03 is 3100. Remember the rule: one digit after the point pairs with ten, two digits pair with a hundred; match the digits to the denominator and the swap never fails.
6. Part 6: Everyday use: money and weightDecimals appear everywhere in daily life. $2.5 means 2 dollars and 50 cents; a weight of 35.5 kg means 35 and a half kilograms; a bottle of juice of 1.2 L is a little more than 1 litre. Whenever a measurement is not a whole number, such as a height of 1.4 m or a temperature of 36.8 degrees, decimals step in to write it. A common mistake is to take $2.5 as 2 dollars and 5 cents; in fact 0.5 of a dollar is 50 cents, and only 0.05 is 5 cents. The first decimal place of money is ten cents and the second is one cent. When reading a price tag, look carefully at where the point sits before working out the amount, and the payment will be right.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• The point separates wholes from parts of one
• 0.1 is one tenth; the first place after the point is tenths
• 0.5 = 510 = one half
• Read 1.8 as one point eight
• Money and measures use decimals when amounts are not whole