Most jobs in New Zealand will require that employees are able to deal with whole numbers. Numbers are used to tell time, count and keep track of money. Nowadays, people need to develop skills based on their ability to work with numbers. Being proficient with whole numbers is the start of this process. For example, how many 'A's are in the following line?
A A A A A A A A AA A A A A A A A A A A A A A
If you got 25, then you got it wrong as the answer is 23. If you got 23 then well done!
This section of the guide will look at aspects of whole numbers and will cover:
Even if you feel reasonably confident with each of these aspects, it might be helpful just to go over the materials as a refresher.
In the decimal system 10 digits are used to represent any whole number. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. These can be arranged to represent any quantity that is required. This is done by placing each digit in place in the number. The position that the digit occupies in the number indicates its value.
The number below is quite large.
5,432,619,278
To investigate its value, we would probably put each digit in table to find out the total value of this number. Each place value to the left is 10 times that on the right.
So, the value of the 9 is 9000.
The value of the 6 is 600,000.
The value of the 8 is 8.
The value of the 4 is 400,000,000.
This number is five billion, four hundred and thirty two million, six hundred and nineteen thousand, and two hundred and seventy eight.
It’d be nice to have this amount in your bank.
From the previous section you can see that place value is very important when we are using numbers. If you get the place value wrong, then you’ll get the number wrong. We can use place value as a way of rewriting numbers to gives a ‘sense’ of how large or small they are. One way of doing this is by using expanded form.
Write the number 345 in expanded form.
Solution
In expanded form the addition would look like this:
3 hundreds + 4 tens + 5 ones
Expanded form for 345 could also be written with brackets and multiplication signs as below.
(3 x 100) + (4 x 10) + (5 x 1)
See the handout in the bottom section for more examples.
Collatz conjecture is also known as the 3x+1 problem. It concerns a series of numbers and whether this series always reaches 1 when starting from any positive whole number. It is also known as the Hailstone Problem.
How it works: Think of a number.
If the number is even, then divide it by 2.
If the number is odd, then multiply it by 3 and add 1.
Keep going.
Example
Let's say the number is 13. This is odd so, multiply by 3 and add 1. This gives 40.
40 is even, so divide by 2. This gives 20.
Keep going. The numbers you get should be:13, 40, 20, 10, 5, 16, 8, 4, 2, 1
Try it with 12. What do you get?
12, 6, 3, 10, 5, 16, 8, 4, 2, 1
Try it with another number. What do you get?
Do you think you'll always get ...4,2,1?
No one has been able to prove that this is the case. Try a few other larger numbers.
For many trade-based jobs, there are times where you may not need an exact mathematical answer for a problem. In these situations, it is useful to be able to estimate an answer. You can do this by rounding off an answer to the nearest 10, or 100, or 1000 etc.
Round the following numbers to the nearest 100.
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a) 728 is closer to 700 than it is to 800, so 728 rounds to 700. b) 791 is closer to 800 than it is to 700, so 791 rounds to 800. See the handout in the bottom section for more examples. |