Decimals

Lesson

Decimals are one way of expressing numbers that are smaller than one. Look at the columns to the right of the decimal point in the place value table below:

These columns give us information about the value of the numbers written in them.

**Evaluate:** What is the place value of the $4$4 in $23.47$23.47?

**Think: **We can see that the $4$4 is in the tenths column, so the place is tenths and the value of the digit is $4$4.

**Do:** We can multiply the place of the digit by the value of the digit to find the place value.

So the place value of the $4$4 is equal to:

$4\times\frac{1}{10}=\frac{4}{10}$4×110=410

We can also combine the place and the value to be written as:

$4$4 tenths

Or we can simply ignore all the other digits except $4$4 and see what its value is. If we do this, we get:

$00.40$00.40

So the place value of the $4$4 is either: $4$4 tenths or $\frac{4}{10}$410 or $0.4$0.4

We removed the unnecessary zeros from $00.40$00.40 to get $0.4$0.4 because removing those zeros did not change the value of the number.

**Evaluate:** What is the place value of the $6$6 in $55.006$55.006?

**Think:** The $6$6 is in the thousandths column so the place is thousandths and the value of the digit is $6$6.

**Do:** We can multiply the place by the value to find the place value.

Similar to the previous example, we can see that the value of the $6$6 can be represented as either:

$6$6 thousandths or $\frac{6}{1000}$61000 or $0.006$0.006

Writing numbers that include decimals in expanded notation is just like what you have done before using whole numbers. You just write the values off the place value table.

**Evaluate: **Write $6384.92$6384.92 in expanded notation

**Think:** Which columns do these numbers belong in on the place value table?

**Do:** We would write this as $6000+300+80+4+\frac{9}{10}+\frac{2}{100}$6000+300+80+4+910+2100

**Evaluate: **Write $1059.004$1059.004 in expanded notation

**Think: **Which columns do these numbers belong in on the place value table?

**Do: **$1000+50+9+\frac{4}{1000}$1000+50+9+41000

We can also use our knowledge of the place value table to change numbers that are written in words to numbers.

**Evaluate:** Write seventeen and four tenths as a decimal

**Think: **The tenths column is the first column after decimal point

Do**:** $17.4$17.4

**Evaluate: **Write six and seventy-seven hundredths as a decimal.

**Think: **$77$77 hundredths will be written in the first two columns after the decimal point

**Do:** $6.77$6.77

We can use this knowledge about the place value table to interpret information about decimals from pictures.

In the picture above, $6$6 out of the $10$10, or $6$6 tenths of the squares are shaded. So in decimal form, we would write this as $0.6$0.6.

In the picture above, $43$43 out of hundred, or $43$43 hundredths of the squares are shaded. So in decimal form, we would write this as $0.43$0.43 .

When we multiply or divide by numbers like $10$10, $100$100 or $1000$1000, we do so by increasing or decreasing the place value of each digit. The number of zeros in the number we are multiplying or dividing by tells us how many places each digit will move. When we are multiplying, the place values will increase and when we are dividing, the place values will decrease.

**Evaluate:** $0.4\times100$0.4×100

**Think:** Multiplying by $100$100 means our number is getting bigger- the digits will increase by two place values.

Hundreds | Tens | Units | . | Tenths | Hundredths |
---|---|---|---|---|---|

$0$0 | . | $4$4 | $0$0 | ||

$0$0 | $4$4 | $0$0 |

**Do: **$0.4\times100=40$0.4×100=40

Notice that we added an extra zero in the Hundredths column so that the units column would not be empty after moving the digits. Similarly, we removed the zero in the Hundreds column since removing it doesn't change the value of the number.

**Evaluate: **$34\div10$34÷10

**Think:** We need to decrease the place value of each digit by one place.

**Do: **$34\div10=3.4$34÷10=3.4

We can use this in everyday contexts as well.

**Evaluate: **Convert $16$16 cents into dollars.

**Think:** There are $100$100 cents in a dollar so we need to divide $16$16 by $100$100, which means we decrease the place value of each digit by two places.

**Do:** $16\div100=\$0.16$16÷100=$0.16

**Evaluate:** Convert $13$13 metres into kilometres.

**Think: **There are $1000$1000 metres in a kilometre so we need to divide $12$12 by $1000$1000, which means we decrease the place value of each digit by three places so that the number is $1000$1000 times smaller.

**Do: **$13\div1000=0.013$13÷1000=0.013km

Understand operations on fractions, decimals, percentages, and integers