Catch up: algebraic operations Flashcards
Distinguish between real, natural and rational numbers and integers. What symbols denote them?
Natural numbers are numbers that are used for counting, 1,2,3. We use the symbol N to denote the set of natural numbers.
Integers are the set of positive and negative whole numbers, …,−3,−2,−1,0,1,2,3,…, which we denote by the symbol Z.
Rational numbers are numbers that can be expressed as a fraction of an integer and a natural number. That is, a rational number r = p
/ q, where the integer p is called the numerator and the natural number q is called the denominator. The fact that p is a natural number is often expressed in symbols as p ∈Z, where ∈ means ‘q is an element of the set of integers’. We denote the set of rational numbers by the symbol Q.
Irrational numbers are numbers that cannot be expressed as a fraction of an integer
and a natural number (e.g pi, sqrt(2) ). The set of irrational numbers is often not given a separate symbol. Instead, the set of real numbers is defined as the combination of the rational and irrational numbers, and is denoted by R.
Is the number 0 included in natural numbers?
the number 0 is not included in N. If we want to include 0, we will use the symbol N0.
What does the requirement q ∈N mean about q?
Note that the requirement q∈N means that q cannot be equal to 0. As we will explain below, division by 0 is not defined.
How are irrational numbers different to rational, natural numbers and integers?
Irrational numbers have no simple real-life correspondence as natural numbers, integers
and rational numbers do. Instead, irrational numbers can be understood as ‘filling in the
gaps’ that are left in the number line by the rational numbers.
Express the relationship between these groups of numbers in symbols
there is a strict inclusion of the sets of numbers introduced above. Each natural number is also an integer, each integer is also a rational number.
In symbols these relations are expressed as N⊂Z⊂Q⊂R, where ‘⊂’ means ‘is a subset of’.
How can real numbers be manipulated? (4)
Real numbers can be
manipulated by four algebraic operations. These are addition, denoted by ‘+’, subtraction, denoted by ‘−’, multiplication, denoted by ‘·’ (or ‘×’), and division, denoted by ‘/’ (or ‘:’).
How do the different operations align with the number groups?
addition is well-defined on the natural numbers (1 + 3 = 4) but subtraction is not (1 - 3 = ?.) This problem is solved by introducing integers, which can also have a negative sign. Similarly, multiplication is well-defined for natural numbers and integers. If Bob forgets his apple three days in a row and borrows an apple from Alice each time, he eventually has 3 ·(−1) = −3 apples. However, the inverse operation, division, is not defined for all integers.
How can these observations of the relatiuon of the operations to the set be described?
the closure property of a set of numbers under an algebraic operation.
Natural numbers are closed under addition and multiplication, that is, if m,n ∈ Nare natural numbers, then also m + n ∈ Nand
m ·n ∈N are natural numbers. The integers are additionally closed under subtraction; if m,n ∈ Zare integers, then also m −n ∈ Zis an integer. Finally, the rational numbers, with the exception of 0, are also closed under division. That is, if r,s ∈Qwith s ̸= 0 (s is not 0), then r/s ∈Qis again a rational number.
The algebraic operations obey certain rules that determine how we carry out computations with real numbers.
Describe this using the correct terminology (5)
Firstly, when using addition and multiplication, the order of the numbers does not matter and we say that addition and multiplication are commutative.
Secondly, when we are applying addition or multiplication to three numbers, the order in which we carry out the operation is irrelevant and we say that addition and multiplication are associative.
Thirdly, when we mix addition and multiplication, the multiplication operation must be applied to each term in the sum separately. This is called the distributive property.
Fourthly, there exist neutral elements for addition multiplication, which have no effect when used in the respective operation. For addition, the neutral element is the number 0; For multiplication, the neutral element is 1.
Finally, subtraction and division are the inverse operations of addition and multiplication. In the case of addition, for every real number we can find a negative number such that their sum equals the neutral element 0. In the case of multiplication, for every non-zero real number r we can find a reciprocal, denoted by r−1, such that the product of the number with its reciprocal equals the neutral element 1, that is, r−1 ·r = r·r−1 = 1.
At the start of this section, we introduced the symbol ‘/’ for the division operation. In the last paragraph, we introduced the notion of a reciprocal.
How are these two notations related?
if a and b are two real numbers, then a/b is shorthand notation for a · b^−1. Due to the commutative property of multiplication, we can exchange the order of a and b−1, that is, a ·b^−1 = b^−1 ·a. However, we cannot exchange the order of the numbers if we are using
the division symbol. Reversing the order of a and b means b/a = b ·a−1; we are now multiplying by the reciprocal of a instead of the reciprocal of b.
To summarize, what is meant by the following terms:
- Closure
- Commutative
- Assosiative
- Distributive
- Neutral Element
- Inverse
Closure: The natural numbers Nare closed under addition and multiplication, the integers Z are additionally closed under subtraction, and rational numbers Q are additionally closed under division.
Commutative: Addition and multiplication are commutative. If x,y ∈ R, then x + y = y + x and x ·y = y ·x
Associative: Addition and multiplication are commutative. If x,y,z ∈ R, then x + (y + z) = (x + y) + z and x ·(y ·z) = (x ·y) ·z.
Distributive When addition and multiplication are mixed, they obey the distributive property. If x,y,z ∈R, then x ·(y + z) = x ·y + x ·z.
Neutral Element The neutral element of addition is 0. For any x ∈ Rit holds x + 0 = x. The neutral element of multiplication is 1. For any x ∈ R it holds x ·1 = x.
Inverse Every real number has an additive inverse. For any x ∈R there is −x ∈R such that x + (−x) = 0. Every non-zero real number has a multiplicative inverse. For any 0 ̸= x ∈R there is x−1 ∈R such that x ·x−1 = 1.
How are rational numbers with equal denominators are added or subtracted?
by adding or subtracting their numerator while leaving the denominator unchanged.
r1 + r2 = p1/ q + p2/ q = p1 ·q^−1 + p2 ·q^−1, which by the distributive property is equal to = (p1 + p2) ·q−1 = p1 + p2 / q
r1 −r2 = r1 + (−r2) = p1 / q + −p2 / q = p1 ·q−1 + (−p2) ·q−1, which by the distributive property is equal to = (p1 + (−p2)) ·q−1 = p1 −p2 / q .
Does (-p)/ q = -(p/q)?
Yes, a property of negative rational numbers is:
If r = p/q ∈Q, then
−p/q = −p/q = p/−q.
(−1)^−1 = −1.
How do you add two rational numbers with unequal denominators?
The simplest solution is to multiply both the numerator and the denominator of r1 by the denominator of r2 and vice versa.
For instance, if r1 = 1/3 and r2 = 2/5, then:
1/3 x 2/5 = 5/15 x 6/15 = 11/15
How can you multiply rational numbers?
Multiplication of rational numbers is carried out separately for the numerator and the denominator. That is, we multiply the numbers above the fraction line and divide them by the product of the
numbers below the fraction line. For instance,
15/2 x 6/7 = 15(6) / 2(7)= 90/14
