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Questions and Answers
Which property of integers is concerned with the order of operands in arithmetic operations?
Which property of integers is concerned with the order of operands in arithmetic operations?
If integers a and b are multiplied, the Distributive Property allows us to:
If integers a and b are multiplied, the Distributive Property allows us to:
When adding two integers of opposite signs, the sum will always be:
When adding two integers of opposite signs, the sum will always be:
Which property ensures the result of an addition operation is the same regardless of the grouping of integers?
Which property ensures the result of an addition operation is the same regardless of the grouping of integers?
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If both integers are positive, what can we conclude about their sum?
If both integers are positive, what can we conclude about their sum?
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Which property does subtraction of integers not follow?
Which property does subtraction of integers not follow?
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What does scalar multiplication involve in terms of changing a vector's size?
What does scalar multiplication involve in terms of changing a vector's size?
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When dividing, what does the divisor represent?
When dividing, what does the divisor represent?
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What relationship do two numbers have if they are unequal?
What relationship do two numbers have if they are unequal?
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If one integer is positive and the other is negative, what property does the result depend on?
If one integer is positive and the other is negative, what property does the result depend on?
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Study Notes
Integers
Integers are whole numbers including positive numbers, negative numbers and zero. They can have any number of digits and extend infinitely to the left (negative) and right (positive). For example, -8, -7, -6, -5...-2, -1, 0, 1, 2, ... 8, 9, 10 and further.
Properties of Integers
Commutativity
The commutative property states that when performing arithmetic operations such as addition or multiplication, we don't really care about the order of the operands. This means that a + b = b + a and a * b = b * a.
Associativity
Associativity is similar to commutativity, but it deals with grouping symbols. It states that regardless of the grouping, the result remains the same. So, (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c).
Distributive Property
The distributive property is another fundamental principle of arithmetic. It states that if you multiply two integers together, you can break up the first integer into individual parts and multiply each part by the second integer separately before adding them all together. For example:
- a(b + c) = ab + ac.
Operations with Integers
Addition
When adding two integers, there are three possible outcomes:
- If both integers are either positive or negative, their sum will also be a member of the original set.
- If one integer is positive and the other negative, the sum will always be a positive number.
- If one integer is negative and the other is positive, the result will depend on the magnitude of the positive integer. For example, -x + y = y - x.
Subtraction
Subtracting one integer from another works similarly to addition. However, unlike addition, subtraction does not follow the commutative property. That is why, when performing subtraction, we need to ensure that the smaller number is subtracted from the larger one.
Multiplication
Multiplication operates with the concept of scalars. In mathematics, scaling can refer to anything we do to transform vectors using some sort of vector space, like addition and subtraction. Scalar multiplication involves changing the size of a vector without causing a change in its direction. In simpler terms, you just multiply the coordinates by a single value.
Division
Division is the inverse operation of multiplication. When dividing, the divisor is the number being divided into groups, and the quotient is the number of groups consisting of that many items. For example, if you want to divide 12 apples equally among 4 people, you would say that the quotient is 3 apples per person.
Comparing Integers
Comparing integers allows us to determine which one is bigger or smaller. There are four types of relationships between integers: equality, less than, greater than, and inequality:
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Equality Two numbers are equal if they represent the same amount.
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Less Than If one number is greater than the other, then we say the larger number is greater than the smaller one.
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Greater Than Two numbers are equal if they represent the same amount.
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Inequality Two numbers are unequal if one is greater than the other and neither is less than the other.
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Description
Learn about integers, whole numbers that include positive numbers, negative numbers, and zero. Explore properties like commutativity, associativity, and the distributive property. Study operations including addition, subtraction, multiplication, and division, as well as comparisons between integers.
