Decimals and place value

Decimals as rational numbers

• Some decimal numbers are rational numbers: but some are not.

• A decimal is a rational number if it can be written as a fraction with integer numerator and denominator. Those are decimals that either terminate (end) or have a repeating block of digits.

• Repeating decimals: 7.6666…; 0.727272…• Terminating decimals: 4.8; 9.00001; 0.75

Irrational numbers• A number that is not rational is called irrational.

• A decimal like 3.5655655565555655556… is not rational because although there is a pattern, it does not repeat. It is an irrational number.

• Compare this to 3.556556556556556556…It is rational because 556 repeats. It is a rational number.

Comparing Decimals

• When are decimals equal?

• 3.56 = 3.56000000

• But, 3.056 ≠ 3.560.

• To see why, examine the place values.

• 3.056 = 3 + 0 • .1 + 5 • .01 + 6 • .001

• 3.560 = 3 + 5 • .1 + 6 • .01 + 0 • .001

• Think of units, rods, flats, and cubes.

Ways to compare decimals

• Write them as fractions and compare the fractions as we did in the last section.

• Use base-10 blocks.

• Use a number line.

• Line up the place values.

Exploration 5.16

• Use the base 10 blocks to represent decimal numbers and justify your answers.

• Work on this together and turn in on Wednesday.

Homework for Wednesday

• Read pp. 308-323 in the textbook

• Exploration 5.16

Rounding

• 3.784: round this to the nearest hundredth.

• 3.784 is between 3.78 and 3.79. On the number line, which one is 3.784 closer to?

• 3.785 is half way in between.

3.78 3.785 3.79

Adding and Subtracting Decimals

• Same idea as with fractions: the denominator (place values) must be common.

• So, 3.46 + 2.09 is really like3 + 2 ones + 4 + 0 tenths + 6 + 9 hundredths = 5.55

Multiplying Decimals• As with whole numbers and fractions,

multiplication of decimals is best illustrated with the area model.

• 2.1 • 1.3 1 + 1 + .11

+

.3

Dividing decimals

• Standard algorithm—why do we do what we do?

Exploration 5.18

• Work on this exploration in class and finish for homework.

• Part 1: 1-4

• Part 2: 1, 2