Decimals

Syllabus tag: KCSE | Mathematics | Form 1 | Topic 8 Decimals

Lesson objectives

By the end of this topic, you should be able to:

• Convert fractions to decimals and decimals to fractions.
• Convert recurring decimals into fractions.
• Round off a decimal to a required number of decimal places.
• Write numbers in standard form.

Decimals

A decimal writes a fraction using place value after a point. Every fraction can be written as a decimal, and every decimal as a fraction.

a) Fractions to decimals

Divide the numerator by the denominator.

Some divisions end. 3 ÷ 8 = 0.375 exactly. These are terminating decimals.

Some never end. 1 ÷ 3 = 0.333… forever. These are recurring decimals.

A fraction terminates when its denominator, in simplest form, has only 2s and 5s as prime factors. Any other prime factor makes it recur.

b) Writing a recurring decimal

Place a dot above the repeating digit, or above the first and last digits of the repeating block.

So 0.333… has a dot over the 3, and 0.454545… has dots over the 4 and the 5.

c) Recurring decimals to fractions

Let x be the decimal.

Multiply by a power of 10 so that the repeating blocks line up. Use 10 for one repeating digit, 100 for two, 1 000 for three.

Subtract the original from the multiplied version. The endless tails cancel exactly, which is the whole trick.

Solve the resulting equation and simplify.

d) Decimals to fractions

For a terminating decimal, write the digits over the matching power of ten, then simplify.

So 0.375 is 375/1000, which simplifies to 3/8.

e) Rounding

Look at the digit after the last place you are keeping.

Five or more rounds up; less than five leaves it. Then drop everything after.

Decimal places are counted after the point. Significant figures are counted from the first non-zero digit.

In 0.004567, the leading zeros only hold the place. The first significant figure is 4.

Always round from the original, never from an already rounded value.

f) Standard form

Standard form writes a number as A × 10ⁿ. The value A is at least 1 and less than 10.

Move the point until exactly one non-zero digit stands before it. Count the places moved.

Moving left gives a positive index. Moving right gives a negative one.

A check: large numbers take a positive index, small numbers a negative one. If 0.00072 came out as 7.2 × 10⁴, the sign is wrong.

g) Where this is used

Money and measurement. Scientific work uses standard form for the size of an atom and the distance to a star. Calculator answers usually need rounding.

Words to know

• Decimal place -- a position after the decimal point.
• Terminating decimal -- one that stops after a finite number of digits.
• Recurring decimal -- one in which a digit or block repeats forever.
• Significant figures -- digits counted from the first non-zero digit.
• Standard form -- A × 10ⁿ where 1 ≤ A < 10 and n is an integer.

:::checkpoint Check yourself

1. Convert 5/8 to a decimal. Does it terminate or recur?
2. Convert 0.2222… to a fraction.
3. Round 0.048372 to 3 significant figures.
4. Write 0.00056 in standard form. :::

Bridge to practice

The exercises begin with place value and converting between fractions and decimals, move through terminating and recurring decimals and the algebraic conversion, and finish with rounding, significant figures and standard form. After writing any answer in standard form, check that A is at least 1 and below 10 before moving on.