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 its position (place) within the number
• Each place has a value ten times larger than the place to the right of it
• e.g.  For the number 9876
  • The 6 represents 6 ones (or units) (6)
  • The 7 represents 7 tens (70)
    
    • "ten" is ten times larger than "one"
  • The 8 represents 8 hundreds (800)
    
    • "hundred" is ten times larger than "ten"
  • The 9 represents 9 thousands (9000)
    
    • "thousand" is ten times larger than "hundred"
  • In words, this number is nine thousand, eight hundred and seventy six

How do I read large numbers?

• Start with the ones (units) digit and work 'right to left' through the digits to deduce the place value that the number starts with 
  • e.g.  For the number 12345678

• • • 12345678 starts in the ten millions place value
    • So it would be read (and written in words) as twelve million, three hundred and forty five thousand, six hundred and seventy eight

How does place value work for decimals?

• Starting with the decimal point
  • digits to the left of the decimal point form the whole number part (ones, tens, thousands, ...)
  • digits to the right of the decimal point form the decimal part
• Each decimal place has a value ten times larger than the place to the right of it
• e.g. For the number 36.952
  • The whole number part is 36 (3 tens and 6 ones)
  • The 9 represents 9 tenths (0.9)
    • "one" is ten times larger than "tenth"
  • The 5 represents 5 hundredths (0.05)
    • "tenth" is ten times larger than "hundredth"
  • The 2 represents 2 thousandths (0.02)
    • "hundredth" is ten times larger than "thousandth"
  • In words, this number is thirty six point nine five two

How do I read decimals?

• The whole number part would be read as above
• The decimal part is read digit by digit
  • e.g.  The number 23.45678 would be read (and written in words) as twenty three point four five six seven eight
• Although they are not read, it is still important to know the value of each decimal place

• • You will often hear these place values used relating to race time
    • e.g.  In a sprint race, athletes may be separated by "five hundredths of a second" (0.05 seconds)

What are negative numbers?

• Negative numbers are any number less than zero
  • They may also be referred to as directed numbers
• Negative numbers are indicated by a minus sign (-)
  • To avoid confusion between subtraction and negative numbers, sometimes the following is used:
    • negative numbers are written in brackets
    • a longer dash is used for subtraction (—)
    • the minus for a negative number is raised (superscript), e.g. ^-4
• Negative numbers are read by using the word 'negative' or 'minus' before the value
  • e.g.  -8 would be read/said as "negative eight" or "minus eight"

What are the rules for working with negative numbers?

• When multiplying and dividing with negative numbers
  • Two numbers with the same sign make a positive
    • 
    • 
  • Two numbers with different signs make a negative
    • 
    • 
• When adding and subtracting with negative numbers
  • Subtracting a negative number is the same as adding the positive
    • e.g. 
  • Adding a negative number is the same as subtracting the positive
    • e.g. 

Where are negative numbers used in real-life?

• Temperature is a common context for negative numbers
  • 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 it as follows:
    • A room is -5°C, then -6°C of cold air is 'removed'
    • The room now warms to 1°C
• Money and debt is another common context for negative numbers
  • A negative sign means you owe money
  • 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 only $100 in debt
    • This is equivalent to (-300) + 200 = -100

How do I put numbers in order (including decimals and negatives)?

• Use place value in the number to help
  • Write the numbers underneath each other, lining up their place-value columns
  • e.g.  For the numbers 1 453 427 and 454 316

• • Start from the highest place value (furthest left) to compare the numbers
• For decimals, numbers further to the right of the decimal point are worth less
  • e.g.  14 is more than 8 but 0.14 is less than 0.8
  • It can help to write the two numbers with the same number of decimal places to compare them
    
    • e.g.  12.115 and 12.15 are easier to compare when written as 12.115 and 12.150
• For negative numbers, larger values are smaller numbers
  • e.g.  14 is more than 8 but -14 is less than -8
  • If there is a mixture, first order the positive numbers and negative numbers separately
• Ascending order means in increasing order
  • Start with the smallest (most negative) number
• Descending order means in decreasing order
  • Start with the largest (most positive) number