Chapter 12: Another Peek Beyond the Point
Decimals are fractions over 10, 100, 1000 …, so we can multiply and divide them. In this chapter we find a quick rule for placing the decimal point in a product, continue long division past the point, divide by decimals, meet divisions that never end, and see how a tiny decimal shaped our leap-year calendar.
In this chapter you will
- convert fractions over powers of 10 to decimals
- multiply decimals and place the point
- divide decimals by whole numbers and by decimals
- explain leap years with decimals
Topics
- A Quick Recap of Decimals NCERT P2 4.1
A fraction over 10, 100, 1000 … is written as a decimal by moving the point left.
- Decimal Multiplication NCERT P2 4.2
Multiply without the points; the product's decimal places = the sum of both numbers' places.
- Decimal Division NCERT P2 4.3
Long division keeps going: regroup ones into tenths, tenths into hundredths, and put the point in the quotient.
- Look Before You Leap! NCERT P2 4.4
A year is 365.2422 days, so the calendar needs leap years to keep up.
Key words
- Decimal place
- a position after the point: tenths, hundredths …
- Regroup
- change 1 one into 10 tenths (and so on) to keep dividing
- Recurring decimal
- a decimal whose digits repeat forever, like 3.333…
- Leap year
- a year with 366 days
Summary
- Multiply decimals without the points, then give the product as many places as both had.
- Divide by 10ⁿ by moving the point left; for a decimal divisor, multiply both numbers by 10ⁿ first.
- Long division regroups into tenths, hundredths …; some divisions repeat forever.
- The year is 365.2422 days, which is why leap years follow the 4–100–400 rule.