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Other number bases: calculations

In this video Michael takes a look at how numbers would be interpreted if we used a base 8 system.
7.3
MICHAEL ANDERSON: Take a look at these two calculations. Both are correct. How can this be so? The answer is that each is using a different number base. The first example is being calculated in base 8, and the second in base 6. Let’s explore further in order to make sense of this. The base 8 system is called octal. The place value grid looks like this. The digits used in base 8 are 0, 1, 2, 3, 4, 5, 6, and 7. There are no 8’s and no 9’s. Once we get to number 8, we write this as 1-0, not 10, which stands for one lot of 8 and no units.
54.6
To write the number we know is 21 in base 8, we have to write 21 as a combination of 8’s. 21 equals 8 plus 8 plus 5. That’s two lots of 8 and 5 left over. So in base 8, we write this as 2-5. We can state this as 21 to the base 10 equals 2-5 to the base 8. From now on, we will write numbers expressed in base 10 as just the number with no suffix. But numbers written in a different base will have a suffix indicating the base. Let’s attempt to write 148 in base 8. We need to construct 148 in chunks of 1, 8, 64, or 512. Two lots of 64 is 128, leaving 20 left over.
111.9
Two lots of 8 is 16, leaving 4 leftover. So 148 is equal to two lots of 64 plus two lots of 8 plus one lot of 4.

Challenge

Take a look at these two calculations. Both are correct when calculated using a base which is not base ten. The base is different for each calculation. Work out which base has been used for each these calculations.
Two calculations are shown in number bases that are not base 10: 37 + 54 = 113 and 124 + 234 = 402
In this video Michael takes a look at how numbers would be interpreted if we used a base 8 system.
In base 8, each place value is a power of 8. The units are represented as \(8^0\), then the next place is not tens, but eights, \(8^1\). Then, the next place is not hundreds as in base 10, but sixty-fours, \(8^2\).
It is common practice to show numbers in different bases by using a subscript suffix. For example, using the results from the calculation above: \(113_8\) and \(402_6\).
You might like to have a go at working out what the numbers in the base 6 example above (right hand calculation) would be in base 8 and base 10?

Problem worksheet

Complete questions eleven 11 and 12 from this week’s worksheet.

Teaching resources

We could easily have used any base systems successfully. It just so happens that we opted for a base ten system, probably because we have ten digits on our hands.
There are a number of resources, useful to explore different number bases in this collection on the STEM Learning website:
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Maths Subject Knowledge: Understanding Numbers

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