The Number System (CIE IGCSE Maths: Core)

What is place value?

• When a number is written down using digits, each digit has a value depending on where it is within the number
• For example for the number 9876
  • The 6 represents 6 ones (or units)
  • The 7 represents 7 tens
  • The 8 represents 8 hundreds
  • The 9 represents 9 thousands
• In words, this number is nine thousand, eight hundred, and seventy six

How do I read large numbers?

• Each place has a value ten times larger than the place to the right of it
• For example for the number 12345678

• When dealing with large numbers, it is often helpful to write them with a space between some of the digits
  • We do this after every 3 digits from the right
  • For example, 12345678 could be written as 12 345 678
• This number would be twelve million, three hundred and forty five thousand, six hundred and seventy eight

How does place value work for decimals?

• Each place has a value ten times larger than the place to the right of it
• This means for decimals, the decimal places further to the right are worth less
• Lets look at the number 23.45678
  • We would read this as twenty three point four five six seven eight
  • However, each place has a different value

• The 4 represents 4 tenths, or 0.4
• The 6 represents 6 thousandths, or 0.006
• You will often hear these place values used relating to time
  • For example in a sprint race, athletes may be separated by "5 hundredths of a second" or 0.05 seconds

What are negative numbers?

• Negative numbers are any number less than zero
  • They may also be referred to as directed numbers
• They appear in lots of places from numerical calculations to algebra 
• You might come across them in real-life problems such as temperature or debt

What are the rules for working with negative numbers?

• When multiplying and dividing with negative numbers
  • Two numbers with the same sign makes a positive
    • e.g and 
  • Two numbers with different signs makes a negative
    • e.g.  and
  • For multiplication and division, it's often easier to calculate ignoring any signs, then making a decision about whether the answer should be positive or negative
• When adding and subtracting with negative numbers
  • Subtracting a negative is the same as adding the positive number
    • e.g.
  • Adding a negative is the same as subtracting the positive number
    • e.g.

Where could negative numbers be used?

• Temperature is a common context for negative numbers, and one that we are used to using
  • If the temperature is 3°C, and it cools by 5°C, the new temperature will be -2°C
    • This is equivalent to 3 - 5 = - 2
  • If the temperature is -4°C, and it warms up by 6°C, the new temperature will be 2°C
    • This is equivalent to (-4) + 6 = 2
  • To explain why (-5) - (-6) = 1, you could think of this as a temperature of -5°C, and then -6°C of cold air is removed, which makes it warmer overall
• Money and debt is another context where negative numbers can be used, where a negative sign represents money that is owed
  • If someone has a debt of $200, and they borrow another $400, their total debt is now $600
    • This is equivalent to (-200) + (-400) = -600
  • If someone is in debt by $300, but then pays off $200 of their debt, they are now $100 in debt
    • This is equivalent to (-300) - (-200) = -100
    • or (-300) + 200 = -100

How do I put numbers in order?

• When comparing two or more numbers, use the place values in the number to help you
• Be careful with decimals, where numbers further to the right of the decimal point are worth less
  • For example 3.14 is more than 3.130
  • It can help to write the two numbers with the same number of decimal places to compare them
    • 3.140 is more than 3.130
• Be careful with negative numbers; where -12 is smaller than -6