Reading and Ordering Numbers

Mathematics (Core) revision notes · Cambridge (CIE) · 0580 · Core

1. Place value

Place value is the value a digit carries because of where it sits in a number
Each column is worth ten times the column immediately to its right
In 5 273, the digits are worth 5 thousands, 2 hundreds, 7 tens and 3 ones
The value of a digit is the digit itself multiplied by its column heading
- The 2 in 5 273 is worth 200, not 2
Large numbers are written with a thin space every three digits, counting leftwards from the ones column
- 41276309 is written 41 276 309
Reading in words follows the same grouping: 41 276 309 is forty-one million, two hundred and seventy-six thousand, three hundred and nine

The decimal point separates two parts: whole units sit to its left, and fractions of a unit to its right
The columns after the point are tenths, hundredths, thousandths, and so on
In 8.407, the 4 is worth 4 tenths, the 0 is worth 0 hundredths and the 7 is worth 7 thousandths
A decimal is read digit by digit after the point: 8.407 is read as eight point four zero seven
Zeros between the point and a non-zero digit are not optional, because they hold the later digits in their columns
- 0.07 and 0.7 differ by a factor of ten

Worked example

Reading a large number

Write 6 208 034 in words, and write down the value of the digit 2.

Solution:

Group the digits in threes from the right: 6 | 208 | 034
The groups are millions, thousands and ones
In words: six million, two hundred and eight thousand and thirty-four
The 2 sits in the hundred-thousands column, so its value is 200 000
The zeros are not optional: they hold the 8 in the thousands column and the 3 in the tens column

Ascending order runs from smallest to largest; descending order runs from largest to smallest
To compare whole numbers, compare the highest place value first, then work rightwards until the digits differ
To compare decimals, give every number the same number of decimal places by padding with zeros
- 3.6, 3.58 and 3.605 become 3.600, 3.580 and 3.605, so the order is 3.58, 3.6, 3.605
Length of a decimal is not a guide to its size: 0.9 is larger than 0.4999

On a number line, values increase from left to right, so any negative number is smaller than any positive number
Among negatives, the further a number is from zero, the smaller its value
- −11 is smaller than −4
With a mixed list, order the negatives and the positives separately, then place all the negatives first

Worked example

Ordering a mixed list of decimals

Write these numbers in ascending order: 0.62, −1.4, 0.7, −0.35, 1.08, −1.09

Solution:

Separate the signs: negatives are −1.4, −0.35, −1.09; positives are 0.62, 0.7, 1.08
Pad the negatives to the same number of decimal places: −1.40, −0.35, −1.09
The further from zero, the smaller the value, so the negatives run −1.40, −1.09, −0.35
Pad the positives: 0.62, 0.70, 1.08, which are already in order
Ascending order: −1.4, −1.09, −0.35, 0.62, 0.7, 1.08

A subtracted negative reverses into an addition
- 8 − (−5) = 8 + 5 = 13
An added negative reverses into a subtraction
- 6 + (−9) = 6 − 9 = −3
Two signs written next to each other collapse to one: two like signs give a plus, two unlike signs give a minus

Matching signs on the two values produce a positive result
- (−7) × (−3) = 21 and (−24) ÷ (−6) = 4
Opposing signs produce a negative result
- (−7) × 3 = −21 and 24 ÷ (−6) = −4
With a longer product, count the negative factors: an even count gives a positive, an odd count gives a negative

Temperature below zero is the most common context
- A temperature of 2 °C falling by 9 °C gives 2 − 9 = −7 °C
Money owed is written as a negative balance
- A balance of −£150 that receives a £90 repayment becomes −150 + 90 = −£60
Depth below sea level and floors below ground are also recorded as negatives

Worked example

Calculations with negative numbers

Work out each of the following. (a) (−6) + 11 (b) 4 − (−7) (c) (−5) × (−8) (d) 36 ÷ (−9)

Solution: