Numbers, Operations, and Absolute Value Study Guide
Introduction to Numbers, Operations, and Absolute Value
Arithmetic is numbers you squeeze from your head to your hand to your pencil to your paper till you get the answer
—Carl Sandburg (1878–1967)
In this lesson, you will discover the ins and outs of integers. Think you already know these math players? This lesson will teach you some shortcuts to operations with integers. You'll also learn facts about zero, absolute value, and number properties.
Numbers, Numbers, Numbers. Before you can begin to build your basic math skills, you'll need to understand the different types of numbers.
Integers are the numbers that you see on a number line.
This does not include fractions, such as and or decimals like 1.125, 2.4, and 9.56. No fractions or decimals are allowed in the world of integers. What a wonderful world, you say. No troublesome fractions and pesky decimals.
Positive integers are integers that are larger than zero. Negative integers are smaller than zero. When you are working with negative and positive integers, try to think about a number line. The following number lines will help you see addition and subtraction from different points on the number line.
Zero is neither positive nor negative.
Even integers can be divided by 2 with no remainder.
The remainder is the number left over after division: 11 divided by 2 is 5, with a remainder of 1.
Even integers include –4,–2, 0, 2, and 4. Odd integers cannot be divided by 2 with no remainder. These would include –3,–1, 1, and 3.
Let's examine the properties of zero:
The sum of any number and zero is that number: 0 + 7 = 7.
The product of any number and zero is zero: 0 × 7 = 0.
If you look at a point on a number line, measure its distance from zero, and consider that value as positive, you have just found the number's absolute value. Let's take the absolute value of 3.
The absolute value of 3, written as |3|, is 3.
Next, let's calculate the absolute value of –3.
The absolute value of –3, written as |–3|, is also 3.
Adding and Subtracting Integers - The Associative and Communative Property
The result in an addition problem is called the sum. The result in a subtraction problem is called the difference.
Adding integers often involves the use of certain properties. The associative property of addition states that when you add a series of numbers, you can regroup the numbers any way you'd like:
1 + (9 + 7) = (1 + 9) + 7 = (1 + 7) + 9
The commutative property of addition states that when you add numbers, order doesn't matter:
8 + 2 = 2 + 8
Multiplying and Dividing Integers
The result in a multiplication problem is called a product. The result in a division problem is called a quotient.
The product of two integers with the same sign (+ and + or – and –) is always positive. The product of two integers with different signs (+ and –) is always negative.
- 3 × 4 = 12
- –3 × –4 = 12
- –3 × 4 = –12
- 3 × –4 = –12
Likewise, the quotient of two integers with the same sign (+ and + or – and –) is always positive. The quotient of two integers with the different signs (+ and –) is always negative.
- 4 ÷ 2 = 2
- –4 ÷ –2 = 2
- –4 ÷ 2 = –2
- 4 ÷ – 2 = –2
When multiplying integers, you will often use the same properties you used with the addition of integers. The associative property of multiplication states that when you are multiplying a series of numbers, you can regroup the numbers any way you'd like:
- 2 × (5 × 9) = (2 × 5) × 9 = (2 × 9) × 5
The commutative property of multiplication states that when you multiply integers, order doesn't matter:
6 × 5 = 5 × 6
Find practice problems and solutions for these concepts at Numbers, Operations, and Absolute Value Practice Questions.
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