Lesson 1, Topic 1
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1.11. Work with rational and irrational numbers

ryanrori January 25, 2021

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When working with rational and irrational numbers, you will encounter algebraic formulae, where letters of the alphabet represent numbers. You will therefore need to ensure that symbols for irrational numbers such as 7c and 42 are left in formulae or steps to calculations except where approximations are required.

Algebraic Expressions

An algebraic expression is a symbolic representation of mathematical operations (+ – x /) that can involve both numbers and variables. An expression is a mathematical statement that may use numbers, variables, or both.  The following are examples of expressions: 

  • 3 + 7 
  • 2 × y + 5 
  • 2 + 6 × (4 – 2) 
  • z + 3 × (8 – z) 
  • John weighs 120 kilograms, and Jack weighs k kilograms. Write an expression for their combined weight. The combined weight in kilograms of these two people is the sum of their weights, which is 120 + k. 

To evaluate an expression at some number means we replace a variable in an expression with the number and simplify the expression. 

Example

  • Evaluate the expression 4 × z + 12 when z = 15. 

We replace each occurrence of z with the number 15, and simplify using the usual rules: parentheses first, then exponents, multiplication and division, then addition and subtraction. 

4 × z + 12 becomes 

4 × 15 + 12 = 60 + 12 = 72 

Note:

When we simplify expressions we can use the following laws:

  • Commutative law 

a + b = b + a 

This means that you can turn the order of the numbers around for addition and multiplication and the answer will be the same.

  • Associative law 

a + (b + c) = (a + b) + c 

This means it does not matter in what order numbers are added or multiplied, the answer will always be the same.

  • Distributive law 

a(b + c) = ab + ac. 

This means that the result of first adding several numbers and then multiplying the sum by some number is the same as first multiplying each separately by the number and then adding the products.

Explore repeating (recurring) decimals

Before you can explore repeating decimals and convert them to common fraction form, you need to understand the concepts of repeating and terminating decimals.

Terminating decimals

The word “terminate” means “end”.

A decimal that ends is a terminating decimal. 

In other words, a terminating decimal doesn’t keep going. A terminating decimal will have a finite number of digits after the decimal point. 

Examples of Terminating Decimals:

http://www.icoachmath.com/images/sitemap/term-dec-1.gif

Repeating (recurring) decimals

A decimal representation of a real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is some finite sequence of digits that is repeated indefinitely. For example, the decimal representation of 1/3 = 0.3333333… (spoken as “0.3 repeating”, or “0.3 recurring”) becomes periodic just after the decimal point, repeating the single-digit sequence “3” infinitely.

Application

How do we know whether a fraction will give a terminating decimal? The rule is to find the prime factors of the denominator.

If the prime factors are only 2 and/or 5 the decimal will terminate.

Examples:

3/28

28 = 2 x 2 x 7

so the decimal will not terminate. It will be a recurring decimal.

7/40

40 = 2 x 2 x 2 x 5

The prime factors of 40 consist of 2s and 5s, so the decimal will terminate.

6/125

125 = 5 x 5 x 5

The decimal will terminate.

71/120

120 = 2 x 2 x 2 x 3 x 5

There is a 3 in there, so the decimal will recur.

Convert to and from fractions

To change a fraction to a decimal, you divide the top number by the bottom number. (Divide the numerator by the denominator.)

Example

To convert 3/8 to a decimal, we calculate 3 ÷ 8

0.375 over 8)3.000

so 3/8 = 0.375 

Some decimals will terminate (end) like the example above, but many will not.

For example 2/7 does not terminate.

2 over 7 = 285714285742... over  2.0000000000000

There are some fraction/decimal equivalents that you should be familiar with:

  • 1/2 = 0.5
  • 1/4 = 0.25
  • 3/4 = 0.75
  • 1/3 = 0.333333… 

Converting a decimal into a fraction

If the decimal terminates (ends), the denominator will be 10, or 100, or 1000 or… (depending on the number of decimal places).

0.5 means ‘five tenths’, so 0.5 = 5/10 = 1/2

0.45 means ’45 hundredths’, so 0.45 = 45/100 = 9/20

0.240 means ‘240 thousandths’, so 0.240 = 240/1000 = 6/25

Etc.

If the decimal repeats with a single digit, the denominator will be 9:

0.2222222… = 2/9

0.4444444… = 4/9

0.6666666… = 6/9 = 2/3

If the decimal repeats with two digits, the denominator will be 99:

0.24242424… = 24/99

Etc.

If the decimal does not repeat at all it is known as an irrational number, and you cannot write it as a fraction.