BINARY NUMBER SYSTEM

Back to: Mathematics Primary 5

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There are various methods of writing, counting and reading numbers. The most popular method is the base ten (denary) system, which is based on the powers of ten (i.e how many rounds of ten).

Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, are used to show numbers in denary (base ten). The position of each digit shows its value. For instance, 4932 means 4 thousand, 9 hundred, 3 tens and 2 units. Apart from base ten system, there are other systems of writing, counting and reading numbers. These include:

• The base nine system, which uses numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, only

 

• The base eight system (octal) which uses numbers 0, 1, 2, 3, 4, 5, 6, 7, only

 

• The base seven system, which uses numbers 0, 1, 2, 3, 4, 5, 6, only

 

• The base six system, which uses numbers 0, 1, 2, 3, 4, 5, only

 

• The base five system, which uses numbers 0, 1, 2, 3, 4, only

 

• The base four system, which uses numbers 0, 1, 2, 3, only

 

• The base three system, which uses numbers 0, 1, 2, only

 

• The base two system (binary), which uses numbers 0 and 1 only

 

Binary operations are therefore operations in base two. The following will be considered under this:

• Changing numbers from base ten to base two

 

• Changing numbers from base two back to base ten

 

• Addition, subtraction and multiplication in base two

 

CHANGING NUMBERS FROM BASE TEN TO BASE TWO

 

We can easily change any number in base ten to a number in base two by dividing the number with 2 and writing out each remainder.

 

Example: Change the following numbers from base ten to binary number

• 55ten
• 46ten
• 103ten

Solution

• 55ten

 

Divisor	Dividend	Remainder
2	55
2	27	1
2	13	1
2	6	1
2	3	0
2	1	1
	0	1

 

Now write the remainders from beneath to the top.

= 110111two

Divisor	Dividend	Remainder
2	46
2	23	0
2	11	1
2	5	1
2	2	1
2	1	0
	0	1

Now write the remainders from beneath up to the top.

= 101110two

 

1. 103ten

 

CHANGING FROM BASE TWO TO BASE TEN

We can easily change a number in base two back to number in base ten by expanding the numbers according to the powers they carry.

 

NB:

20=1

2¹=2×1=2

2²=2×2=4

2³=2×2×2=8

2⁴=2×2×2×2=16

Examples: Change the following numbers from binary to base ten.

• 110111two

 

• 101110two

 

• 1100111two

 

Solution

• 1⁵1⁴ 0³1² 1¹1⁰two

=(1×2⁵)+(1×2⁴)+(0×2³)+(1×2²)+(1×2¹)+(1×2⁰)

128

 

• (1×32)+(1×16)+(0×8)+(1×4)+(1×1)
• 32+16+0+4+1
• 53

 

• 1⁵0⁴1³1²1¹0⁰two
• (1×2⁵)+(0×2⁴)+(1×2³)+(1×2²)+(1×2¹)+(0×2⁰)
• (1×32)+(0×16)+(1×8)+(1×4)+(1×2)+(0×1)
• 32+0+8+4+2+0
• 46

 

• 1⁶1⁵0⁴0³1²1¹1⁰two
• (1×2⁶)+(1×2⁵)+(0×2⁴)+(0×2³)+(1×2²)+(1×2¹)+(1×2⁰)
• (1×64)+(1×32)+(0×16)+(0×32)+(1×4)+(1×2)+(1×1)
• 64+32+0+0+4+2+1
• 103

 

 

MULTIPLICATION OF NUMBERS IN BASE TWO

Examples: find the product of the following numbers

• 1102 × 112

 

• 11112 × 112
• 1012 × 1012

Solution

• 1102

1 12

 

1 1 0

 

• 1 1 0

 

1001 02

• 11112

1 12

 

1111

 

• 1111

 

	1011012
c.	1012

× 1012

 

1 0 1

 

1 0 1

 

+101

 

1000112

 

 

 

Quiz

• Change the following numbers to binary:

 

• 75

 

• 63

 

• 54

 

• 82

 

• 143

 

• Change the following binary numbers to a number in base ten:

 

• 111112

 

• 110012

 

• 101112

 

• 1011102

 

• 1000002

 

• Simplify the following in base two:

 

• 10102–1012

 

110012–11112

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