MATH 1F – Mathematics in the Modern World
001 – Real Number System and Imaginary Numbers
Real Number System
 REAL NUMBER is a number that can be
found on the number line. These are the
numbers that we normally use and apply in
real-world applications.
 TERMINATING DECIMALS
o 0.035
 NON-TERMINATING RECURRING
DECIMALS
o 0.555…
 NON-TERMINATING NON-RECURRING
DECIMALS
o 1.4142…
Convert the following Decimals numbers to
Fractions.
 “Real” does not mean they are in the real
world
o In mathematics, when we write 0.5,
we mean exactly half.
o But in the real-world half may not be
exact
o They are not called "Real" because
they show the value of something
real.
TERMINATING DECIMALS
REAL NUMBERS ( R )
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MATH 1F – Mathematics in the Modern World
001 – Real Number System and Imaginary Numbers
NON-TERMINATING RECURRING DECIMALS
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MATH 1F – Mathematics in the Modern World
001 – Real Number System and Imaginary Numbers
NON-TERMINATING NON-RECURRING
DECIMALS
Phases of Imaginary numbers
𝒊 - √−𝟏
𝟐
𝒊𝟐 - 𝒊 ∗ 𝒊 = √−𝟏 ∗ √−𝟏 = (√−𝟏) = −𝟏
𝒊𝟑 - 𝒊𝟐 ∗ 𝒊 = −𝟏(√−𝟏) = −√−𝟏
𝒊𝟒 - 𝒊𝟐 ∗ 𝒊𝟐 = −𝟏(−𝟏) = 𝟏
NOT REAL NUMBERS
 Imaginary Numbers (i)
o √−1 (the square root of minus 1)
 Infinity
 Exponents with multiples of 4 will lead to a
value of 1.
o In excess of 1 will mean i.
o 2 will mean i2, and
o 3 will mean i3.
Imaginary numbers
 An imaginary number is a complex
number that can be written as a real
number multiplied by the imaginary
unit i, which is defined by its property i2 =
−1
 Originally coined in the 17th century
by René Descartes as a derogatory term
and regarded as fictitious or useless,
 the concept gained wide acceptance
following the work of Leonhard Euler (in the
18th century) and Augustin-Louis
Cauchy and Carl Friedrich Gauss (in the
early 19th century).
 An imaginary number bi can be added to a
real number a to form a complex number of
the form a + bi,
 where the real numbers a and b are called,
respectively,
 the real part and the imaginary part of the
complex number.
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