Problem 50 - Algebra 1: Concepts and Skills

Question

Write the numbers in order from least to greatest. (Skills Review p. 770) $$1301,1103,1031,1013,1130$$

Step-by-Step Solution

Answer

The numbers in ascending order are: 1013, 1031, 1103, 1130, 1301.

Step 1 Title

Start by comparing the first digit of each number. The numbers that start with '1' are smaller than those that start with any other number. All the given numbers have '1' as the first digit.

Step 2 Title

Next, look at the second digit of each number. The lower the second digit, the smaller the number. Comparing the second digits, the numbers '1031' and '1013' have the smallest second digit which is '0'.

Step 3 Title

Comparing the third digits of the numbers '1031' and '1013' which are '3' and '1' respectively, the number with the lesser third digit is smaller. Hence, '1013' is the smallest number.

Step 4 Title

Next smallest number can be found by comparing the remaining numbers i.e., '1301','1103','1031', and '1130'. Looking at the second digits we see that '1031' is the next smallest number.

Step 5 Title

Finally, comparing the second digits for the remaining numbers i.e., '1301', '1103' and '1130', '1103' is the next smallest number. And amongst '1301' and '1130', '1130' is smaller.

Step 6 Title

Therefore, the numbers in ascending order are '1013', '1031', '1103', '1130', '1301'

Key Concepts

Comparing Digits

When we compare digits to determine the order of numbers, we start at the leftmost digit and move right. This approach is crucial because in our base-10 number system, the leftmost digits have the greatest value. For multi-digit numbers starting with the same digit, the next digit is then compared.
For example, in comparing the numbers 1031 and 1013, we see that both begin with '1'. Moving to the second digit, both are '0', so we compare the third digit where 1031 has a '3' and 1013 has a '1'. Since '1' is less than '3', we conclude 1013 is the smaller number. An analogy would be arranging words alphabetically; the position of each letter in the word plays a role in its order just like the digits in a number. We always examine digit by digit from left to right, ensuring we clearly understand the value of each before making a decision.

Ascending Order

Arranging numbers in ascending order means listing them from the smallest to the largest. It's similar to climbing a ladder where each step takes you a bit higher. To arrange numbers in ascending order, we can use the comparing digits method we just discussed.
Once the smallest number is determined, it's set aside as the starting point of the sequence. The process is then repeated for the remaining numbers, incrementally finding the next smallest until the largest number caps the sequence. This orderly approach streamlines the task and reduces errors. It's like sorting a hand of playing cards from lowest to highest; each card is examined and placed in its proper position relative to the others.

Number Sequence

A number sequence is a list of numbers arranged according to a certain rule. When we talk about sequences in math, it's not just any random collection of numbers, but an ordered set that follows specific patterns or rules. Ordering numbers from least to greatest, as we did in the exercise, creates a specific type of sequence known as an ascending sequence.
Sequences can be all sorts of things, such as increasing by the same amount (like the counting numbers 1, 2, 3, ...), decreasing, or even following more complex patterns like the Fibonacci sequence. Understanding the underlying rule of a sequence allows us to predict future numbers in the pattern and even find numbers that might be missing. In daily life, sequences can be found in schedules, timelines, and the arrangement of items according to size or importance.