Decimals

Syllabus tag: Kenya CBC | Grade 8 Mathematics | Strand 1.0 Numbers | Sub-Strand 1.3 Decimals (8 lessons)

Lesson objectives

By the end of this sub-strand, you should be able to:

• Convert fractions to decimals, and identify recurring decimals
• Convert recurring decimals into fractions
• Round off a decimal number to a required number of decimal places or significant figures
• Express numbers in standard form
• Carry out combined operations on decimals

Decimals

A decimal is another way of writing a fraction, using place value after a point. This topic converts between the two and handles very large and very small numbers.

a) Fraction to decimal

Divide the top by the bottom.

Some divisions stop. 3 ÷ 4 gives exactly 0.75. These are terminating decimals.

Some never stop. 1 ÷ 3 gives 0.333… going on forever. These are recurring decimals.

A recurring decimal is written with a dot over the repeating digit. For a repeating block, put dots over its first and last digits.

b) Which fractions terminate

A fraction terminates if its denominator, in simplest form, has only 2s and 5s as prime factors.

So 1/8 terminates, because 8 is 2 × 2 × 2. And 1/6 recurs, because 6 contains a 3.

c) Recurring decimal to fraction

Some recurring decimals are worth knowing by sight. 0.333… is 1/3, 0.666… is 2/3, and 0.111… is 1/9.

For others there is a method. Call the decimal x. Multiply by a power of 10 so the repeating part lines up, subtract, then solve.

For 0.444…, let x = 0.444…. Then 10x = 4.444…. Subtracting gives 9x = 4, so x = 4/9.

d) Rounding

Decimal places are counted after the point.

Significant figures are counted from the first non-zero digit.

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

Look at the next digit to decide. Five or more rounds up, less than five leaves it alone.

Always round from the original number, never from an already rounded one.

e) 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 only one non-zero digit sits in front of it. Count how many places you moved.

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

Check: a big number has a positive power, a small one a negative power. If you have written 0.0067 as 6.7 × 10³ you have the sign the wrong way round.

f) Combined operations

Work in the usual order. Brackets first, then multiply and divide, then add and subtract.

Line up the decimal points when adding or subtracting.

When multiplying, count the total decimal places in both numbers and give the answer that many.

g) Where this is used

Money is decimal. Measurements are decimal. Scientists use standard form for the size of an atom and the distance to a star. The ordinary way would need dozens of zeros.

Words to know

• Recurring decimal -- a decimal in which one or more digits repeat forever, shown with a dot or bar over the repeating part.
• Significant figures -- the digits in a number that carry meaning for its precision, counted from the first non-zero digit.
• Standard form -- writing a number as a value between 1 and 10 multiplied by a power of 10.

:::checkpoint Check yourself

1. Convert 5/8 to a decimal. Does it terminate or recur?
2. Round 3.14159 to 3 decimal places and to 3 significant figures.
3. Write 62 000 in standard form.
4. Write 0.00045 in standard form. :::

Bridge to practice

Try the exercises below -- on converting fractions to decimals, recurring decimals, rounding, significant figures, and standard form.