Place value and ordering
Every column is worth ten of the next
In 4,372.18 the 3 is worth 300, because it sits in the hundreds column.
| Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths |
|---|---|---|---|---|---|---|
| 4 | 3 | 7 | 2 | . | 1 | 8 |
| 4000 | 300 | 70 | 2 | 0.1 | 0.08 |
The value of a digit is the digit times its column's value. The 8 sits in the hundredths column, so it is worth 8 × 0.01 = 0.08.
The digits after the decimal point are its decimal places: 4,372.18 has 2 decimal places.
Multiplying and dividing by 10, 100 and 1000
Moving one column left multiplies a digit's value by 10.
Moving one column right divides it by 10.
Multiplying by 100 moves every digit two places left: 3.47 × 100 = 347. Dividing by 1000 moves them three places right: 52 ÷ 1000 = 0.052.
Multiplying by 0.1 is the same as dividing by 10, and dividing by 0.01 is the same as multiplying by 100.
- 52 × 0.1 = 52 ÷ 10 = 5.2
- 0.3 ÷ 0.01 = 0.3 × 100 = 30
Ordering decimals: line up the points
Which is bigger, 0.6 or 0.59? The second has more digits, but that does not matter. Line up the decimal points and fill gaps with zeros:
- 0.600
- 0.590
- compare the tenths first: 6 is more than 5
- so 0.6 is bigger than 0.59
Ordering negative numbers
With negative numbers, the further from zero, the smaller: −7 is less than −4.
A number line helps: left is smaller.
The same goes for negative decimals: −0.45 is further from zero than −0.4, so −0.45 is smaller.
Gaps and halfway points on the number line
The difference between two temperatures is the distance between them on the number line.
From −5 °C up to 9 °C: count 5 degrees up to zero, then 9 more, so the difference is 14 degrees. The lowest temperature is the most negative one.
The number halfway between two numbers is found by adding them and halving the total.
- halfway between −3.4 and 1
- add them: −3.4 + 1 = −2.4 (start at −3.4 and move 1 to the right)
- halve the total: −2.4 ÷ 2 = −1.2
- check: −3.4 to −1.2 is a gap of 2.2, and −1.2 to 1 is also 2.2
The symbols
- = equals, ≠ does not equal
- < less than, > greater than (the wide end faces the bigger number)
- ≤ less than or equal to, ≥ greater than or equal to
An integer is a whole number: positive, negative or zero (…, −2, −1, 0, 1, 2, …).
So −2 ≤ n < 3 includes −2 but not 3. If n is an integer, n can be −2, −1, 0, 1, 2.
With < at both ends, as in −3 < n < 1, neither end is included: n can be −2, −1, 0.
To choose a symbol between two numbers, write both the same way and compare column by column. 0.08 and 0.080 have the same digits in every column, so 0.08 = 0.080.
The most common slip in ordering: treating 0.125 as bigger than 0.5 because 125 is bigger than 5. Always compare column by column.
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