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📐 Mathematics - Class 7
Lesson 12 · Whole Numbers

Binary and Base Five Numbers

MathematicsSubject
14 minEstimated read
Ten is a choice, not a law

We write numbers in groups of ten, and it feels like the only possible way, but it is a habit that comes from having ten fingers. Nothing in mathematics requires it.

What base ten actually means

In the number 347, each column is worth ten times the one to its right: 3 lots of 100, plus 4 lots of 10, plus 7 lots of 1.

Change the ten and everything follows

If the columns were worth five times the one to the right, the place values would be 1, 5, 25, 125, and you would only need the digits 0 to 4. The base is simply how many you count before carrying.

Base two, or binary

Place value 168421
Digits for 1301101
What it adds to08401

Thirteen is written 1101 in base two, because 8 plus 4 plus 1 is 13. Only the digits 0 and 1 are ever used, because as soon as you reach the base, you carry.

Say the digits, do not read it as a number

The binary number 1101 is not one thousand one hundred and one. Read it as one, one, zero, one in base two, or say thirteen if you want its value.

Converting from base ten

Divide the number by the base, note the remainder, divide the answer by the base again, and carry on until you reach zero. Then read the remainders UPWARDS.

Why the remainders go upwards

The first remainder belongs in the ones column, the rightmost place. Each later remainder belongs one column further left. So the last remainder you write is the leftmost digit, which is why you read from the bottom of your working to the top.

Converting back to base ten

Write the place values above the digits, multiply each digit by its place value, and add. For 1101 in base two the value is 8 + 4 + 0 + 1 = 13. For 243 in base five it is 50 + 20 + 3 = 73.

Why computers use base two

A wire can be carrying current or not, and a switch can be on or off. Two states are exactly what base two needs. Building a machine that reliably distinguishes ten voltage levels would be far harder than one that only tells on from off.

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Worked Example 1

Convert 45 from base ten into base two.

Divide repeatedly by 245, 22, 11, 5, 2, 1, 0 with remainders 1, 0, 1, 1, 0, 1.
Read upwards45 = 101101 in base two.
Check by converting back32 + 8 + 4 + 1 = 45. It agrees.
Worked Example 2

Convert 73 into base five, and 243 from base five back into base ten.

73 into base fiveRemainders 3, 4, 2, read upwards: 243 in base five.
243 back into base ten2 × 25 + 4 × 5 + 3 × 1 = 73. The two conversions agree.
Notice the digits usedIf your base five answer contains a 5 or a 6, you have made a mistake: those digits do not exist in that base.
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Activity 12.4

Count to thirty on five fingers using binary

The ideaIn binary each finger is worth a different place value, so one hand reaches much further than five.
MethodThumb 1, then 2, 4, 8 and 16. A finger up counts, a finger down does not. Add the fingers that are up.
Then count in orderCount from zero up to thirty one one at a time, writing the binary digits as you go.
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1

Base ten is a habit that comes from having ten fingers, not a rule of mathematics.

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1Explain what the base of a number system means, using base ten and base five as examples.
2Convert 45 into base two and explain why the remainders are read upwards.
3Why does base two use only the digits 0 and 1, and why do computers use it?
4Add 1011 and 110 in base two without converting, and explain how carrying works.
5How high can you count on ten fingers using binary, and why is it so much more than ten?

Question 1 of 10

1In base ten, each column is worth
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Slide 1 of 4
Unit 12 · Base Systems

Base Systems

Mathematics · Grade 7

What You Will Be Able To Do

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Explain what base ten actually means, in terms of place value

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Convert any number into base two or base five by repeated division

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Convert back using place values, which also checks your first answer

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Say why every computer in the world stores numbers in base two

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We write numbers in groups of ten, and it feels like the only possible way, but it is a habit that comes from having ten fingers. Nothing in mathematics requires it.

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Presenter notes: Open with a question nobody has been asked before. Write 347 on the board and ask why we group numbers in tens. Most students have never considered that there might be a reason, and the honest answer is slightly deflating: because we have ten fingers. Nothing in mathematics requires it. Hold up both hands and say that if humans had eight fingers, every number in the room would be written differently and none of them would have changed value.