Chapter Notes: Number, Decimal Numbers

Decimal Numbers

Decimal numbers include a whole number part and a fractional part separated by a decimal point.
Example: The number 1.25 is read as "one point two five" and consists of:

• 1 one (whole number part).
• 2 tenths and 5 hundredths (fractional part).

Tenths are 10 times smaller than ones:

• Example: 1/10 = 0.1.
• 2 tenths = 0.2.

Hundredths are 100 times smaller than ones:

• Example: 1/100 = 0.05.
• 5 hundredths = 0.05.

Representation of 1.25:

• 1 whole + 2 tenths + 5 hundredths = 1 + 0.2 + 0.05 = 1.25.
• Visually, 2 tenths are 2 of 10 equal rows, and 5 hundredths are 5 of 100 equal squares.

Equivalence to fractions:

• 0.25 = 25/100 = 1/4 (since 25 hundredths simplifies to one quarter).
• 1.25 = 125/100 = 5/4 (since 1 whole = 100 hundredths, so 1.25 = 100 + 25 = 125 hundredths).

Reading and writing decimal numbers:

• 75.3: Seventy-five point three.
• 7.35: Seven point three five.
• 0.75: Zero point seven five.
• 90.48: Ninety point four eight.
• 94.08: Ninety-four point zero eight.
• 940.8: Nine hundred forty point eight.
• The word "zero" is used to clarify digits of 0 in the ones or decimal positions (e.g., 0.75 is "zero point seven five").

Place Value

Decimal numbers can be decomposed into place value parts:

• Example: 369.2 = 3 hundreds + 6 tens + 9 ones + 2 tenths.
• Example: 36.92 = 3 tens + 6 ones + 9 tenths + 2 hundredths.
• Example: 234,045 = 2 hundred thousands + 3 ten thousands + 4 thousands + 0 hundreds + 4 tens + 5 ones.
• Example: 23,404.5 = 2 ten thousands + 3 thousands + 4 hundreds + 0 tens + 4 ones + 5 tenths.
• Example: 340.45 = 3 hundreds + 4 tens + 0 ones + 4 tenths + 5 hundredths.

Regrouping allows numbers to be expressed differently:

• Example: 15.35 = 15 ones + 35 hundredths.
• Example: 15.35 = 10 ones + 5 ones + 3 tenths + 5 hundredths.

Using counters to represent numbers:

• Counters in place value positions (e.g., tens, ones, tenths) show the number's decomposition.
• Regrouping counters can show the same number in different ways (e.g., 10 tenths = 1 one).

Rounding to the Nearest Whole Number

Rounding to the nearest whole number involves identifying the previous and next whole numbers.\|
Rules for rounding:

If the tenths digit is less than 5, round down to the previous whole number.

• Example: 4.1 rounds to 4.

If the tenths digit is 5 or greater, round up to the next whole number.

• Example: 4.8 rounds to 5.
• Example: 23.5 rounds to 24.
• A number with 5 tenths (e.g., 3.5) rounds up to the next whole number.

Multiplying and Dividing Decimals by 10 and 100

Multiplying a decimal by 10 shifts digits one place to the left:

• Example: 5.3 × 10 = 53.
• Example: 5.38 × 10 = 53.8.

Dividing a decimal by 10 shifts digits one place to the right:

• Example: 53 ÷ 10 = 5.3.
• Example: 53.8 ÷ 10 = 5.38.

Multiplying by 100 shifts digits two places to the left:

• Example: 4.8 × 100 = 480.

Dividing by 100 shifts digits two places to the right:

• Example: 48 ÷ 100 = 0.48.
• Example: 5.38 ÷ 100 = 0.0538.

Relationships between masses or heights:

• Example: If a truck's load is 775.3 kg and a van's load is 77.53 kg, the truck's mass is 775.3 ÷ 77.53 = 10 times larger.

Patterns and Sequences

A linear sequence has a constant difference between consecutive terms.
Term-to-term rule: Add or subtract the constant difference to get the next term.

Example: Sequence of straws (30, 18, 7, ...):

• Differences: 30 - 18 = 12, 18 - 7 = 11 (inconsistent, suggesting a different rule or error).
• If corrected to a linear sequence like 30, 26, 22, 18, the rule is subtract 4.

Finding terms:

• Example: For a sequence with term-to-term rule add 4, starting at 18, the sixth term is 18 + 4 × 4 = 34 (two jumps of 4 from the fourth term).