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Essential Concepts

• Counting numbers are the ones we use to count objects, like [latex]1[/latex], [latex]2[/latex], [latex]3[/latex], and so on. When we add zero to the counting numbers, we get a new set of numbers called whole numbers.
• In a number, each digit’s position has a value, and a place value chart helps us understand this. The chart groups the values into periods like ones, thousands, millions, and so on, and each place is worth ten times the value of the place to its right, similar to how base- [latex]10[/latex] blocks work.
• To name a whole number using place value, start from the left and say each period’s number followed by its name, without using “s” at the end. Use commas to separate the periods. When writing a number given in words, identify the periods, draw blanks for the places, and place the digits in the correct positions based on their place value.
• Rounding whole numbers means making them simpler by adjusting them to the closest specified value. To round a number, find the given place value, look at the digit to the right, and if it’s [latex]5[/latex] or greater, add [latex]1[/latex] to the digit in the given place value; if it’s less than [latex]5[/latex], leave the digit unchanged and replace all digits to the right with zeros.
• We can perform different mathematical operations on whole numbers. We learned how to add, subtract, multiply, and divide them. We also learned how to write these operations using notation, how to read them, and what the results mean.
  • In addition, we have two important properties. The Identity Property of Addition says that when we add any number to zero, the result is the same number. The Commutative Property of Addition tells us that changing the order of the numbers we add does not change the sum.
  • In multiplication, we have a few important properties. The Multiplication Property of Zero tells us that anything multiplied by zero is always zero. The Identity Property of Multiplication says that when we multiply any number by one, the result is the same number. And just like with addition, the Commutative Property of Multiplication tells us that changing the order of the numbers we multiply does not change the product.
  • In division, we have some important properties. The Division Property of One tells us that any number divided by itself is always [latex]1[/latex], and any number divided by [latex]1[/latex] is itself. However, division by zero is undefined, meaning we cannot divide any number by zero. Dividing zero by any number (except zero) always gives us zero.
• When we talk about the opposite of a number, we mean a number that is the same distance from zero on the number line but on the other side. For example, the opposite of [latex]3[/latex] is [latex]-3[/latex].
• Integers are whole numbers, their opposites, and zero, like [latex]-3[/latex], [latex]-2[/latex], [latex]-1[/latex], [latex]0[/latex], [latex]1[/latex], [latex]2[/latex], [latex]3[/latex], and so on.
• On the number line, positive numbers increase in value as we move to the right, and negative numbers decrease in value as we move to the left. We can compare and order positive and negative numbers using inequality symbols, like “<” (less than) and “>” (greater than). For example, if a number is to the left of another number on the number line, it is less than that number, and if it is to the right, it is greater than that number.
• Absolute value is a way to measure how far a number is from zero on the number line. We write the absolute value of a number as two straight lines around it, like |n|. The important thing to know is that absolute value is always positive or zero, it can never be negative.
• Similarly to operations on whole numbers, we have operations on integers. We can add, subtract, multiply and divide integers. There are a few things to note.
  • When you add positive and negative numbers with the same sign, you add their values. For example, when you add [latex]5[/latex] and [latex]3[/latex] (both positive), the sum is positive. But when you add numbers with different signs, you look at which sign has more. If there are more negatives, the sum is negative, and if there are more positives, the sum is positive.
  • When you multiply two numbers with the same sign (both positive or both negative), the product is always positive. But when you multiply numbers with different signs (one positive and one negative), the product is always negative. Also, when you multiply a number by [latex]-1[/latex], it gives you the opposite of that number. For example, [latex]-1[/latex] multiplied by a is equal to -a.
  • When you divide two numbers with the same sign (both positive or both negative), the quotient is always positive. However, when you divide numbers with different signs (one positive and one negative), the quotient is always negative. Also, when you divide a number by [latex]-1[/latex], it gives you the opposite of that number. For example, a divided by [latex]-1[/latex] is equal to -a.
• The order of operations (PEMDAS) tells us which operations to perform first.
  • P – Parentheses: Start by solving any operations inside parentheses first.
  • E – Exponents: Next, simplify any exponents or powers.
  • MD – Multiplication and Division: Then, perform any multiplication or division from left to right.
  • AS – Addition and Subtraction: Lastly, complete any addition or subtraction from left to right.

Glossary

absolute value

the absolute value of a number is its distance from [latex]0[/latex] on the number line

addends

numbers being added in an addition operation

counting numbers

start with [latex]1[/latex] and continue

difference

the result of subtracting

dividend

the number being divided

divisor

the number dividing a number

expression

math statement that includes numbers and operations

factor

number being multiplied

integers

the set of counting numbers, their opposites, and [latex]0[/latex]

opposites

the number that is the same distance from zero on the number line but on the opposite side of zero

partial product

separate product of the digits a number is being multiplied by

periods

place values that are separated into groups of three

place value system

when the value of a digit depends on its position, or place, in a number

product

the result of multiplying

quotient

the result of dividing

remainder

number left over from a division operation

rounding

a method used to shorten or simplify numbers by adjusting them to the closest specified value

sum

the result of addition

whole numbers

the counting numbers and zero